Conditional probability: occupied boxes and sampling without replacement

1. Audio
   Observation

   Narration paraphrase: Event A: When placing 4 distinct balls into 4 distinct boxes, at least one box is empty.

2. Formula
   Observation

   The capital letter A is written below the problem statement on the screen and used as an event symbol in subsequent formulas.

A

Event A: When placing 4 distinct balls into 4 distinct boxes, at least one box is empty.

Sample space consisting of all possible distributions of 4 distinct balls into 4 distinct boxes (classical probability model).

1. Audio
   Observation

   Narration paraphrase: Event B: When placing 4 distinct balls into 4 distinct boxes, exactly two boxes are empty.

2. Formula
   Observation

   The capital letter B is written below the problem statement on the screen and used as an event symbol in the conditional probability formula.

B

Event B: When placing 4 distinct balls into 4 distinct boxes, exactly two boxes are empty.

Sample space consisting of all possible distributions of 4 distinct balls into 4 distinct boxes (classical probability model).

1. Audio
   Observation

   Narration paraphrase: The conditional probability of event B occurring given that event A has occurred.

2. Formula
   Observation

   P(B|A) is written on the screen.

P(B|A)

The conditional probability of event B occurring given that event A has occurred.

Requires $P(A)>0$; in this problem, A is "at least one box is empty".

1. Audio
   Observation

   Narration paraphrase: The probability that events A and B occur simultaneously, i.e., the probability of the intersection event AB.

2. Formula
   Observation

   P(AB) is written in the numerator position of the fraction on the screen.

P(AB)

The probability that events A and B occur simultaneously, i.e., the probability of the intersection event AB.

A and B are events in the same probability space.

1. Audio
   Observation

   Narration paraphrase: The probability of event A, i.e., the probability that at least one box is empty.

2. Formula
   Observation

   $P(A)$ is written in the denominator position of the fraction on the screen, followed by a new line writing $P(A)$=.

$P(A)$

The probability of event A, i.e., the probability that at least one box is empty.

A is "at least one box is empty".

1. Audio
   Observation

   Narration paraphrase: The total number of ways to distribute 4 distinct balls into 4 distinct boxes.

2. Formula
   Observation

   $4^4$ is written on the screen and serves as the denominator for the expression of $P(A)$.

$4^4$

The total number of ways to distribute 4 distinct balls into 4 distinct boxes.

Each ball independently chooses one of the 4 boxes; empty boxes are allowed.

1. Audio
   Observation

   Narration paraphrase: The number of ways to arrange 4 distinct balls into 4 distinct boxes such that no box is empty.

2. Formula
   Observation

   $A_4^4$ is written on the screen and located in the numerator position of the fraction 1 - ...

1. {"text": "The handwritten subscripts and superscripts are small, but combined with the audio mentioning 'permutations and combinations' and 'four balls exactly placed in four boxes', it can be identified as $A_4^4$."}

$A_4^4$

The number of ways to arrange 4 distinct balls into 4 distinct boxes such that no box is empty.

The number of permutations of 4 distinct elements into 4 distinct positions.

1. Formula
   Observation

   The letter A is annotated below the problem statement "at least one box is empty" on the screen.

2. Audio
   Observation

   Narration paraphrase: Event "after placing 4 distinct balls into 4 distinct boxes, at least one box is empty".

A

Event "after placing 4 distinct balls into 4 distinct boxes, at least one box is empty".

Classical probability model where the sample space consists of all ways to place 4 distinct balls into 4 distinct boxes.

1. Formula
   Observation

   The letter B is annotated below the problem statement "exactly two boxes are empty" on the screen.

2. Audio
   Observation

   Narration paraphrase: Event "after placing 4 distinct balls into 4 distinct boxes, exactly two boxes are empty".

B

Event "after placing 4 distinct balls into 4 distinct boxes, exactly two boxes are empty".

Sub-event within the same sample space.

1. Formula
   Observation

   The bottom left of the screen shows P(B|A)=P(AB)/P(A).

2. Audio
   Observation

   Narration paraphrase: Conditional probability of event B occurring given that event A has occurred.

P(B|A)

Conditional probability of event B occurring given that event A has occurred.

Probability value, range [0,1].

1. Formula
   Observation

   P(AB) appears on the screen and is later written as P(AB)=$P(B)$.

2. Audio
   Observation

   Narration paraphrase: Probability of the intersection event where both event A and event B occur.

P(AB)

Probability of the intersection event where both event A and event B occur.

Probability value.

1. Formula
   Observation

   The top right of the screen shows $P(A)=1-A_4^4/4^4$, with the numerator expanded as $4\times 3\times 2$.

2. Audio
   Observation

   Narration paraphrase: Number of permutations of taking 4 elements from 4 distinct elements, i.e., 4!.

$A_4^4$

Number of permutations of taking 4 elements from 4 distinct elements, i.e., 4!.

Non-negative integer count.

1. Audio
   Observation

   Narration paraphrase: The problem requires finding the probability of "exactly two boxes being empty" given the condition "at least one box is empty". The instructor first defines the condition event as A and the target event as B, then writes the required probability as P(B|A), using the conditional probability formula to transform it into P(AB)/P(A).

2. Formula
   Observation

   P(B|A)=P(AB)/P(A) is written on the screen.

The problem requires finding the probability of "exactly two boxes being empty" given the condition "at least one box is empty". The instructor first defines the condition event as A and the target event as B, then writes the required probability as P(B|A), using the conditional probability formula to transform it into P(AB)/P(A).

1. A is "at least one box is empty".

2. B is "exactly two boxes are empty".

3. Using this formula requires $P(A)>0$.

1. Defining Events A and B

1. Audio
   Observation

   Narration paraphrase: To translate the textual conditions into probability symbols, the instructor defines A as representing "at least one box is empty" and B as representing "exactly two boxes are empty". This allows the conditional probability in the original problem to be written as P(B|A).

2. Formula
   Observation

   A and B are labeled below the problem statement on the screen.

To translate the textual conditions into probability symbols, the instructor defines A as representing "at least one box is empty" and B as representing "exactly two boxes are empty". This allows the conditional probability in the original problem to be written as P(B|A).

1. The experiment involves placing 4 distinct balls into 4 distinct boxes.

2. A focuses on the number of empty boxes being at least 1.

3. B focuses on the number of empty boxes being exactly 2.

1. Audio
   Observation

   Narration paraphrase: This problem treats the placement of each ball as an independent choice. Since the 4 balls are distinct and the 4 boxes are distinct, each ball has 4 box options, resulting in a total number of distributions of $4\times 4\times 4\times 4=4^4$. This number serves as the denominator for probability calculations in the classical model.

2. Formula
   Observation

   $4^4$ is written on the screen as the denominator.

This problem treats the placement of each ball as an independent choice. Since the 4 balls are distinct and the 4 boxes are distinct, each ball has 4 box options, resulting in a total number of distributions of $4\times 4\times 4\times 4=4^4$. This number serves as the denominator for probability calculations in the classical model.

1. The balls are distinct from each other.

2. The boxes are distinct from each other.

3. Each ball must be placed into a box.

4. Empty boxes are allowed.

1. Conditional Probability Problem: 4 Distinct Balls into 4 Distinct Boxes

1. Audio
   Observation

   Narration paraphrase: Event A is "at least one box is empty". Directly enumerating cases with 1, 2, 3, or 4 empty boxes is cumbersome. The instructor uses the complementary event instead: no box is empty. Since there are 4 balls and 4 boxes, having no empty boxes means each box contains exactly one ball, and the number of such distributions is $A_4^4$. Therefore, $P(A)=1-P(A^c)=1-\frac{A_4^4}{4^4}$.

2. Formula
   Observation

   $P(A)=1-\frac{A_4^4}{4^4}$ is written on the screen.

1. {"text": "The audio starts with 'at least one box is not empty', but the problem statement and subsequent formulas point to 'at least one box is empty'. Here, based on the visual formula and context, it is understood as a slip of the tongue or transcription error."}

Event A is "at least one box is empty". Directly enumerating cases with 1, 2, 3, or 4 empty boxes is cumbersome. The instructor uses the complementary event instead: no box is empty. Since there are 4 balls and 4 boxes, having no empty boxes means each box contains exactly one ball, and the number of such distributions is $A_4^4$. Therefore, $P(A)=1-P(A^c)=1-\frac{A_4^4}{4^4}$.

1. The total number of outcomes in the sample space is $4^4$.

2. $A^c$ represents "no box is empty".

3. When 4 distinct balls are placed into 4 distinct boxes with no empty boxes, each box has exactly one ball.

1. Defining Events A and B
2. Total Number of Basic Events in Classical Model is $4^4$

1. Formula
   Observation

   The bottom left of the screen clearly writes P(B|A)=P(AB)/P(A).

2. Audio
   Observation

   Narration paraphrase: This problem asks for the probability of "exactly two boxes being empty" under the condition that "at least one box is empty", so the definition of conditional probability is used, writing the target as P(B|A)=P(AB)/P(A).

This problem asks for the probability of "exactly two boxes being empty" under the condition that "at least one box is empty", so the definition of conditional probability is used, writing the target as P(B|A)=P(AB)/P(A).

1. $P(A)>0$.

2. A and B belong to the same sample space.

1. P(B|A)
2. P(AB)
3. A
4. B

1. Formula
   Observation

   The denominators for both $P(A)$ and $P(B)$ on the screen are written as $4^4$.

2. Audio
   Observation

   Narration paraphrase: When placing 4 distinct balls one by one into 4 distinct boxes, each ball has 4 choices, so the total number of arrangements is $4^4$. This is the common denominator for all probability calculations in this problem.

When placing 4 distinct balls one by one into 4 distinct boxes, each ball has 4 choices, so the total number of arrangements is $4^4$. This is the common denominator for all probability calculations in this problem.

1. The balls are distinct from each other.

2. The boxes are distinct from each other.

3. Each ball must be placed into exactly one box.

1. $4^4$

1. Formula
   Observation

   The top right of the screen writes $P(A)=1-A_4^4/4^4$.

2. Audio
   Observation

   Narration paraphrase: Event A is "at least one box is empty". Its complement event is "no box is empty", meaning all 4 boxes are non-empty. Since the number of balls equals the number of boxes and both are distinct, this is equivalent to placing exactly 1 ball in each box, which has $A_4^4$ arrangements. Thus, $P(A)=1-A_4^4/4^4$.

Event A is "at least one box is empty". Its complement event is "no box is empty", meaning all 4 boxes are non-empty. Since the number of balls equals the number of boxes and both are distinct, this is equivalent to placing exactly 1 ball in each box, which has $A_4^4$ arrangements. Thus, $P(A)=1-A_4^4/4^4$.

1. Applicable to "at least one" type events, often calculated via the complement event "none".

1. A
2. $A_4^4$
3. $4^4$
4. Total Count for Classical Probability Model of Balls into Boxes

1. Audio
   Observation

   Narration paraphrase: If B⊆A, then A∩$B=B$, therefore P(AB)=$P(B)$. In this problem, "exactly two boxes are empty" necessarily implies "at least one box is empty", so the numerator P(AB) can be directly replaced with $P(B)$.

2. Formula
   Observation

   The bottom left of the screen writes P(AB)=$P(B)$.

If B⊆A, then A∩$B=B$, therefore P(AB)=$P(B)$. In this problem, "exactly two boxes are empty" necessarily implies "at least one box is empty", so the numerator P(AB) can be directly replaced with $P(B)$.

1. Must first determine the subset relation between events; cannot assume any two events satisfy this simplification.

1. A
2. B
3. P(AB)

1. Formula
   Observation

   The screen writes $P(B)= [C_4^2(C_4^3 C_2^1 + C_4^2)] / 4^4$.

2. Audio
   Observation

   Narration paraphrase: To have exactly two empty boxes, first from 4 boxes choose 2 to be empty, which has $C_4^2$ ways; then place the 4 distinct balls into the remaining 2 non-empty boxes, ensuring neither is empty. Based on the distribution of balls in the two boxes, there are only two types: $3+1$ type and $2+2$ type. The $3+1$ type corresponds to $C_4^3 C_2^1$; the $2+2$ type corresponds to $C_4^2$, but the video has not yet finished writing the subsequent correction term within this clip.

1. At the end of the clip, the instructor just said "but don't forget", whether subsequent steps involve dividing by 2 or other corrections were not provided within this clip.

To have exactly two empty boxes, first from 4 boxes choose 2 to be empty, which has $C_4^2$ ways; then place the 4 distinct balls into the remaining 2 non-empty boxes, ensuring neither is empty. Based on the distribution of balls in the two boxes, there are only two types: $3+1$ type and $2+2$ type. The $3+1$ type corresponds to $C_4^3 C_2^1$; the $2+2$ type corresponds to $C_4^2$, but the video has not yet finished writing the subsequent correction term within this clip.

1. Balls are distinct, boxes are distinct.

2. "Exactly two boxes are empty" means the other two boxes must both be non-empty.

1. B
2. $C_4^2$
3. $C_4^3$
4. $C_2^1$
5. $C_4^2$
6. Total Count for Classical Probability Model of Balls into Boxes

1. Formula
   Observation

   The top-left of the screen writes P(B|A)=P(AB)/P(A).

2. Audio
   Observation

   Narration paraphrase: This problem uses the definition of conditional probability: finding the probability of event B given that event A has already occurred is equal to the probability of the intersection event AB divided by the probability of A.

This problem uses the definition of conditional probability: finding the probability of event B given that event A has already occurred is equal to the probability of the intersection event AB divided by the probability of A.

1. $P(A)>0$.

2. A and B are events in the same probability space.

1. Formula
   Observation

   The screen writes both P(AB)=$P(B)$ and $P(A)$ in the form of 'number of favorable outcomes / $4^4$'.

2. Audio
   Observation

   Narration paraphrase: In an equiprobable sample space, the probability of an event can be calculated as 'number of sample points satisfying the condition / total number of sample points'; the total number of sample points in this problem is $4^4$.

In an equiprobable sample space, the probability of an event can be calculated as 'number of sample points satisfying the condition / total number of sample points'; the total number of sample points in this problem is $4^4$.

1. Each basic outcome is equally likely.

2. The sample space is finite.

1. Conditional Probability Formula

1. Caption evidence
   Observation

   Narration paraphrase: Let A be 'at least one box is empty' and B be 'exactly two boxes are empty'. Since 'exactly two boxes are empty' necessarily implies 'at least one box is empty', we have B⊆A, and consequently AB=B.

2. Formula
   Observation

   The screen uses A to label 'at least one box is empty' and B to label 'exactly two boxes are empty'.

Let A be 'at least one box is empty' and B be 'exactly two boxes are empty'. Since 'exactly two boxes are empty' necessarily implies 'at least one box is empty', we have B⊆A, and consequently AB=B.

1. The 4 balls are distinct.

2. The 4 boxes are distinct.

3. Each ball is placed into one box, resulting in a total of $4^4$ placements.

1. Conditional Probability Formula

1. Audio
   Observation

   Narration paraphrase: In the experiment of placing 4 distinct balls into 4 distinct boxes, if A represents "at least one box is empty" and B represents "exactly two boxes are empty", then B⊆A, and therefore AB=B.

2. Formula
   Observation

   P(B|A)=P(AB)/P(A) appears on the screen.

In the experiment of placing 4 distinct balls into 4 distinct boxes, if A represents "at least one box is empty" and B represents "exactly two boxes are empty", then B⊆A, and therefore AB=B.

1. A: At least one box is empty.

2. B: Exactly two boxes are empty.

3. The sample space of the same experiment consists of all $4^4$ distributions.

For all distributions satisfying B, the number of empty boxes is 2, which necessarily satisfies the condition that the number of empty boxes is at least 1.

1. Audio
   Observation

   Narration paraphrase: When placing 4 distinct balls into 4 distinct boxes with no empty boxes, the number of distributions is $A_4^4$.

2. Formula
   Observation

   $A_4^4$ is written on the screen.

When placing 4 distinct balls into 4 distinct boxes with no empty boxes, the number of distributions is $A_4^4$.

1. The balls are distinct.

2. The boxes are distinct.

3. Each box has at least one ball.

4. The total number of balls equals the total number of boxes, both being 4.

For all distributions with no empty boxes, they correspond to a one-to-one mapping of 4 distinct balls to 4 distinct boxes.

1. Audio
   Observation

   Narration paraphrase: In the sample space of "placing 4 distinct balls into 4 distinct boxes", if B represents "exactly two boxes are empty" and A represents "at least one box is empty", then B⊆A, thus P(AB)=$P(B)$.

2. Formula
   Observation

   Subsequently writes P(AB)=$P(B)$.

In the sample space of "placing 4 distinct balls into 4 distinct boxes", if B represents "exactly two boxes are empty" and A represents "at least one box is empty", then B⊆A, thus P(AB)=$P(B)$.

1. A is defined as at least one box being empty.

2. B is defined as exactly two boxes being empty.

Holds for all arrangements in the sample space of this problem.

1. Formula
   Observation

   Actual board temporarily displays3/32 along the complementary calculation. Editorial clarification: this is the no-empty-box probability; the source later writes1 minus3/32=$29/32$ for $P(A)$.

2. Audio
   Observation

   Narration paraphrase: Editorial verification: $P(A)=29/32$;$3/32$ is the probability of no empty box. The intermediate board temporarily shows $3/32$. At 249–258 seconds the source subsequently writes 1 minus $3/32$, giving $29/32$; the intermediate label is not the final conclusion.

Editorial verification: $P(A)=29/32$;$3/32$ is the probability of no empty box. The intermediate board temporarily shows $3/32$. At 249–258 seconds the source subsequently writes 1 minus $3/32$, giving $29/32$; the intermediate label is not the final conclusion.

1. Placing 4 distinct balls into 4 distinct boxes, total arrangements is $4^4$.

2. A is at least one box being empty.

For the specific experiment given in this problem.

1. Formula
   Observation

   The screen writes P(AB)=$P(B)$.

2. Caption evidence
   Observation

   Narration paraphrase: If B represents 'exactly two boxes are empty' and A represents 'at least one box is empty', then B⊆A, therefore AB=B, and thus P(AB)=$P(B)$.

If B represents 'exactly two boxes are empty' and A represents 'at least one box is empty', then B⊆A, therefore AB=B, and thus P(AB)=$P(B)$.

1. The sample space consists of placing 4 distinct balls into 4 distinct boxes.

2. A and B are defined as above.

Holds for all box placement outcomes satisfying the problem conditions.

1. Formula
   Observation

   The screen finally writes P(B|A)=$\frac{21}{58}$.

2. Audio
   Observation

   Narration paraphrase: Given that at least one box is empty, the probability that exactly two boxes are empty is $21/58$.

Given that at least one box is empty, the probability that exactly two boxes are empty is $21/58$.

1. Placing 4 distinct balls into 4 distinct boxes.

2. A = at least one box is empty, B = exactly two boxes are empty.

The unique numerical result for this specific problem setup.

1. Audio
   Observation

   Narration paraphrase: The probability that the second draw is defective given the first draw was defective is P(B|A)=P(AB)/P(A).

2. Formula
   Observation

   Handwritten P(B|A)=P(AB)/P(A).

The probability that the second draw is defective given the first draw was defective is P(B|A)=P(AB)/P(A).

1. A is the event that the first draw is defective

2. B is the event that the second draw is defective

3. $P(A)>0$

Holds for the specific case of drawing two items without replacement as given in the problem.

1. Audio
   Observation

   Narration paraphrase: The required conditional probability equals $3/19$, corresponding to option C.

2. Formula
   Observation

   Handwritten final result $3/19$, corresponding to option C.

The required conditional probability equals $3/19$, corresponding to option C.

1. There are 16 good items and 4 defective items

2. 2 items are drawn sequentially without replacement

3. It is known that the first draw was defective

Specific to the sampling problem stated in the question.

1. Audio
   Observation

   Narration paraphrase: The original problem's requirement is reduced to calculating $\frac{P(AB)}{P(A)}$, with the next step being to find $P(A)$.

2. Formula
   Observation

   A and B are labeled on the screen first, then P(B|A)=P(AB)/P(A) is written.

1. Expression

   $$$$
   Explanation

   Identify the condition and target parts of the problem: the condition is "at least one box is empty", and the target is "exactly two boxes are empty" under this condition.

   Justification

   The original problem text is "then the probability that exactly two boxes are empty given that at least one box is empty".

   Shown in the video
2. Expression

   $$A=\{\text{至少一个盒子为空}\},\quad B=\{\text{恰好两个盒子为空}\}$$
   Explanation

   Introduce event symbols A and B to convert natural language events into set events.

   Justification

   The instructor explicitly says "this is event A" and "this is event B".

   Shown in the video
3. Expression

   $$P(B\mid A)$$
   Explanation

   The required probability is the probability of B occurring given that A has occurred.

   Justification

   The definition of conditional probability denotes "B happening under condition A" as P(B|A).

   Shown in the video
4. Expression

   $$P(B\mid A)=\frac{P(AB)}{P(A)}$$
   Explanation

   Use the conditional probability formula to transform the target into the ratio of the intersection probability to the condition probability.

   Justification

   The instructor says "according to our formula, it is P(AB) divided by $P(A)$".

   Shown in the video

The original problem's requirement is reduced to calculating $\frac{P(AB)}{P(A)}$, with the next step being to find $P(A)$.

1. Audio
   Observation

   Narration paraphrase: By the end of the clip, $P(A)=1-\frac{A_4^4}{4^4}$ is obtained, but the calculation of P(AB) or the final conditional probability has not yet continued.

2. Formula
   Observation

   $P(A)=1-\frac{A_4^4}{4^4}$ is written on the screen.

1. {"text": "The audio phrase 'at least one box is not empty' contradicts the problem text 'at least one box is empty'; the visual formula supports that it should be 'at least one box is empty'."}

1. Expression

   $$|\Omega|=4^4$$
   Explanation

   Determine the size of the sample space: each ball has 4 box options, and there are 4 balls.

   Justification

   The instructor says "for each ball there should be four choices, so it is four to the power of four".

   Shown in the video
2. Expression

   $$A^c=\{\text{没有盒子为空}\}$$
   Explanation

   Take the complementary event of A, which is that there are no empty boxes.

   Justification

   The instructor says "no box is empty is the complementary event of at least one box is empty".

   Shown in the video
3. Expression

   $$|A^c|=A_4^4$$
   Explanation

   When there are no empty boxes, the 4 distinct balls must enter the 4 distinct boxes respectively, equivalent to permuting the 4 boxes.

   Justification

   The instructor says "four balls exactly placed in four boxes, that implies A44".

   Shown in the video
4. Expression

   $$P(A^c)=\frac{A_4^4}{4^4}$$
   Explanation

   In the classical probability model, the probability of an event is the number of favorable basic events divided by the total number of basic events.

   Justification

   The instructor states "this is a classical probability model" and places $A_4^4$ in the numerator and $4^4$ in the denominator of the fraction.

   Shown in the video
5. Expression

   $$P(A)=1-P(A^c)=1-\frac{A_4^4}{4^4}$$
   Explanation

   Obtain $P(A)$ from the relationship of complementary event probabilities.

   Justification

   The instructor explicitly uses "complementary event" and writes the fractional form 1-...

   Shown in the video

By the end of the clip, $P(A)=1-\frac{A_4^4}{4^4}$ is obtained, but the calculation of P(AB) or the final conditional probability has not yet continued.

1. Formula
   Observation

   Actual board temporarily displays3/32 along the complementary calculation. Editorial clarification: this is the no-empty-box probability; the source later writes1 minus3/32=$29/32$ for $P(A)$.

2. Audio
   Observation

   Narration paraphrase: Site verification: $P(A)=29/32$. The original video temporarily writes the complement fraction in this segment; subsequent supplementation steps and the final answer are correct.

1. Expression

   $$P(A)=1-\frac{A_4^4}{4^4}$$
   Explanation

   Use the complement event to find the probability of "at least one box is empty".

   Justification

   The complement event is "no box is empty", i.e., all 4 boxes are non-empty.

   Shown in the video
2. Expression

   $$A_4^4=4\times3\times2$$
   Explanation

   Expand the permutation number.

   Justification

   Placing 4 distinct balls into 4 distinct boxes with exactly one ball per box is equivalent to the full permutation of 4 balls.

   Shown in the video
3. Expression

   $$4^4=4\times4\times4\times4$$
   Explanation

   Expand the total number of arrangements.

   Justification

   Each ball has 4 independent choices.

   Shown in the video
4. Expression

   $$P(A)=1-\frac{4\times3\times2}{4\times4\times4\times4}=1-\frac{3}{32}=\frac{29}{32}$$
   Explanation

   Site distinguishes between the complement event probability $3/32$ and the conditional event probability $29/32$.

   Justification

   Independent enumeration of 256 assignments gives 24 with no empty box and 232 with at least one. The source at 249–258 seconds subsequently gives the same complement value; at 266 seconds its final $21/58$ agrees.

   Supplementary explanation

Site verification: $P(A)=29/32$. The original video temporarily writes the complement fraction in this segment; subsequent supplementation steps and the final answer are correct.

1. Audio
   Observation

   Narration paraphrase: The numerator in this problem can be changed to $P(B)$, so P(B|A)=$P(B)/P(A)$.

2. Formula
   Observation

   The screen writes P(AB)=$P(B)$.

1. Expression

   $$P(B|A)=\frac{P(AB)}{P(A)}$$
   Explanation

   First write out the definition of conditional probability.

   Justification

   Conditional probability formula.

   Shown in the video
2. Expression

   $$B\subseteq A$$
   Explanation

   Determine event relationship: exactly two empty boxes necessarily leads to at least one empty box.

   Justification

   Directly derived from event definitions.

   Shown in the video
3. Expression

   $$P(AB)=P(B)$$
   Explanation

   The intersection event equals the contained event.

   Justification

   If B⊆A, then A∩$B=B$.

   Shown in the video

The numerator in this problem can be changed to $P(B)$, so P(B|A)=$P(B)/P(A)$.

1. Formula
   Observation

   The numerator of $P(B)$ on the screen is gradually written as $C_4^2(C_4^3 C_2^1 + C_4^2)$.

2. Audio
   Observation

   Narration paraphrase: Within the clip, it can be confirmed that the counting for $P(B)$ starts from $C_4^2(C_4^3C_2^1+C_4^2)$, but the final complete expression is not given.

1. The clip ends at "but don't forget", without seeing whether subsequent steps involve dividing by 2 or other corrections.

2. Therefore, this derivation can only confirm up to "starting to establish the counting expression for $P(B)$", and cannot confirm the final complete expression.

1. Expression

   $$P(B)=\frac{\#B}{4^4}$$
   Explanation

   First write the probability of "exactly two boxes are empty" as a classical probability ratio.

   Justification

   Total number of arrangements is $4^4$.

   Shown in the video
2. Expression

   $$\#B=C_4^2\times(\text{把4个不同球放入2个指定非空盒且均非空的放法数})$$
   Explanation

   First select which two boxes are empty.

   Justification

   From 4 distinct boxes select 2 empty boxes, giving $C_4^2$ choices.

   Shown in the video
3. Expression

   $$3+1\text{ 型}:\ C_4^3C_2^1$$
   Explanation

   One type of distribution is putting 3 balls in one box and 1 ball in the other.

   Justification

   From 4 balls choose 3 as a group. From 2 occupied boxes choose 1 for these 3 balls; the remaining 1 ball enters the other box.

   Shown in the video
4. Expression

   $$2+2\text{ 型}:\ C_4^2\ (\text{片段内尚未写完修正})$$
   Explanation

   Another type of distribution is putting 2 balls in each of the two boxes.

   Justification

   From 4 balls choose 2 as one group. The narration continues with $C_2^2$ and a counting caution; its continuation lies in the next adjacent segment.

   Shown in the video

Within the clip, it can be confirmed that the counting for $P(B)$ starts from $C_4^2(C_4^3C_2^1+C_4^2)$, but the final complete expression is not given.

1. Formula
   Observation

   The screen sequentially writes P(B|A)=P(AB)/P(A), P(AB)=$P(B)=C_4^2(C_4^3C_2^1+C_4^2)/4^4$, $P(A)=1-A_4^4/4^4$, and simplifies to $21/58$.

2. Audio
   Observation

   Narration paraphrase: The original video's final answer of $21/58$ is correct; after reviewing the original frame'$s C_2^1$, this site corrected the model's OCR misreading and its self-derived erroneous arithmetic, without attributing the model's misreading to the original author.

1. This site verified $C_2^1$ against the original frame and corrected the model's recognition; this is not an error by the original author.

1. Expression

   $$P(B\mid A)=\frac{P(AB)}{P(A)}$$
   Explanation

   First, write down the conditional probability formula.

   Justification

   Definition of conditional probability.

   Shown in the video
2. Expression

   $$B\subseteq A\Rightarrow AB=B$$
   Explanation

   'Exactly two boxes are empty' necessarily belongs to 'at least one box is empty'.

   Justification

   From the subset relationship of events, the intersection event equals the smaller event.

   Derived from the video
3. Expression

   $$P(AB)=P(B)=\frac{C_4^2(C_4^3C_2^1+C_4^2)}{4^4}$$
   Explanation

   Use classical probability for B: first select 2 empty boxes, then use the remaining 2 occupied boxes to distribute 4 distinct balls.

   Justification

   Total placements are $4^4$; the numerator is the counting formula given on the screen.

   Shown in the video
4. Expression

   $$C_4^2=6,\quad C_4^3C_2^1=4\times2=8,\quad C_4^2=6$$
   Explanation

   The actual factor chooses one of two boxes:4 times 2 gives 8.

   Justification

   The original frame at 266 seconds shows $C_2^1$ and 4 times 2; independent counting also yields 8 ways for the $3+1$ distribution.

   Supplementary explanation
5. Expression

   $$P(B)=\frac{6\times(8+6)}{4^4}=\frac{6\times14}{256}$$
   Explanation

   For each pair of boxes, there are $8+6=14$ surjective distributions.

   Justification

   The actual blackboard writing is 4 times 2 plus 6; this site corrects the 16 plus 6 derived by the model from OCR misreading.

   Supplementary explanation
6. Expression

   $$\frac{6\times14}{256}=\frac{84}{256}=\frac{21}{64}$$
   Explanation

   There are 84 equiprobable configurations for the target event, giving a probability of $21/64$.

   Justification

   Independent enumeration confirms 84 configurations with exactly two empty boxes; the original blackboard writing is also $21/64$.

   Supplementary explanation
7. Expression

   $$P(A)=1-\frac{A_4^4}{4^4}=1-\frac{24}{256}=1-\frac{3}{32}=\frac{29}{32}$$
   Explanation

   Use the complementary event to find the probability of 'at least one empty box'.

   Justification

   All non-empty is equivalent to one ball per box, totaling 4! ways.

   Shown in the video
8. Expression

   $$P(B\mid A)=\frac{21/64}{29/32}=\frac{21}{64}\times\frac{32}{29}=\frac{21}{58}$$
   Explanation

   Dividing the correct numerator by the conditional probability yields $21/58$, consistent with the original video.

   Justification

   Independent enumeration gives $84/232=21/58$; the original frame at 266 seconds shows the same conclusion.

   Supplementary explanation

The original video's final answer of $21/58$ is correct; after reviewing the original frame'$s C_2^1$, this site corrected the model's OCR misreading and its self-derived erroneous arithmetic, without attributing the model's misreading to the original author.

1. Caption evidence
   Observation

   Narration paraphrase: If the second problem is solved according to the statement 'draw 2 items sequentially without replacement, given the first is defective, find the probability the second is also defective', the answer is $3/19$, corresponding to option C; however, this clip only shows the problem, not the derivation.

2. Audio
   Observation

   Narration paraphrase: If the second problem is solved according to the statement 'draw 2 items sequentially without replacement, given the first is defective, find the probability the second is also defective', the answer is $3/19$, corresponding to option C; however, this clip only shows the problem, not the derivation.

1. The clip ends immediately after displaying the problem, without showing the solution process.

1. Expression

   $$N=16+4=20$$
   Explanation

   The total number of products is 20.

   Justification

   The problem statement gives 16 genuine items and 4 defective items.

   Shown in the video
2. Expression

   $$D_1:\ \text{第一次摸到次品},\quad D_2:\ \text{第二次摸到次品}$$
   Explanation

   Let $D_1$ and $D_2$ represent drawing a defective item on the first and second draws, respectively.

   Justification

   Given $D_1$, the problem asks for the probability of $D_2$.

   Derived from the video
3. Expression

   $$P(D_2\mid D_1)=\frac{P(D_1D_2)}{P(D_1)}$$
   Explanation

   Write down the conditional probability formula.

   Justification

   Definition of conditional probability.

   Derived from the video
4. Expression

   $$P(D_1)=\frac{4}{20}$$
   Explanation

   On the first draw, from 20 products there are 4 defective items that may be drawn.

   Justification

   Classical probability.

   Derived from the video
5. Expression

   $$P(D_1D_2)=\frac{4}{20}\cdot\frac{3}{19}$$
   Explanation

   After drawing a defective item first, 19 items remain, of which 3 are defective.

   Justification

   Sampling without replacement.

   Derived from the video
6. Expression

   $$P(D_2\mid D_1)=\frac{\frac{4}{20}\cdot\frac{3}{19}}{\frac{4}{20}}=\frac{3}{19}$$
   Explanation

   Cancel out $P(D_1)$ to obtain the conditional probability.

   Justification

   Algebraic simplification.

   Derived from the video

If the second problem is solved according to the statement 'draw 2 items sequentially without replacement, given the first is defective, find the probability the second is also defective', the answer is $3/19$, corresponding to option C; however, this clip only shows the problem, not the derivation.

1. Audio
   Observation

   Narration paraphrase: Using the conditional probability formula route yields P(B|A)=$3/19$.

2. Formula
   Observation

   Handwritten P(B|A)=P(AB)/P(A), P(AB)=$4/20\times 3/19$, $P(A)=4/20$, finally writing $3/19$.

1. Expression

   $$P(B\mid A)=\frac{P(AB)}{P(A)}$$
   Explanation

   First translate the problem statement "second draw defective given first draw defective" into conditional probability, and apply the conditional probability formula.

   Justification

   Conditional probability formula directly provided in the video.

   Shown in the video
2. Expression

   $$P(AB)=\frac{4}{20}\times\frac{3}{19}$$
   Explanation

   Calculate the joint probability of both draws being defective: first draw picks from 20 items one of 4 defectives, second draw picks from the remaining 19 items one of the remaining 3 defectives under no-replacement condition.

   Justification

   Step-by-step multiplication under sampling without replacement; video audio and board work show $4/20$ and $3/19$.

   Shown in the video
3. Expression

   $$P(A)=\frac{4}{20}$$
   Explanation

   Calculate the probability that the first draw is defective, i.e., picking from 20 items one of 4 defectives.

   Justification

   Classical probability for a single draw; video directly writes $4/20$.

   Shown in the video
4. Expression

   $$P(B\mid A)=\frac{\frac{4}{20}\times\frac{3}{19}}{\frac{4}{20}}=\frac{3}{19}$$
   Explanation

   Substitute P(AB) and $P(A)$ into the conditional probability formula; the common factor $4/20$ cancels out, yielding the final result $3/19$.

   Justification

   Algebraic cancellation: the source cancels a fraction with denominator 20 and numerator 4, equivalently denominator 5 and numerator 1.

   Shown in the video

Using the conditional probability formula route yields P(B|A)=$3/19$.

1. Audio
   Observation

   Narration paraphrase: Directly deriving from the state after the condition occurs gives P(B|A)=$3/19$, without needing to calculate joint probability first and then divide.

2. Formula
   Observation

   Handwritten $3/19$ on the right side.

1. Expression

   $$\text{Given }A$$
   Explanation

   Treat "first draw is defective" as an established fact, rather than part of the probability to be calculated.

   Justification

   Condition given in the problem statement.

   Supplementary explanation
2. Expression

   $$N_{\mathrm{remaining}}=20-1=19$$
   Explanation

   Because sampling is without replacement, removing one item leaves 19 items in total.

   Justification

   Definition of sampling without replacement.

   Supplementary explanation
3. Expression

   $$D_{\mathrm{remaining}}=4-1=3$$
   Explanation

   Since it is known that the first item removed was defective, the number of remaining defectives decreases from 4 to 3.

   Justification

   Condition A occurred and the extracted item was defective.

   Supplementary explanation
4. Expression

   $$P(B\mid A)=\frac{3}{19}$$
   Explanation

   In the new reduced sample space, the second draw being defective means picking from the 19 remaining items one of the 3 defectives.

   Justification

   Classical probability on the reduced sample space.

   Shown in the video

Directly deriving from the state after the condition occurs gives P(B|A)=$3/19$, without needing to calculate joint probability first and then divide.

1. Caption evidence
   Observation

   Narration paraphrase: Place 4 distinct balls into 4 distinct boxes. What is the probability that exactly two boxes are empty given that at least one box is empty?

2. Audio
   Observation

   Narration paraphrase: Place 4 distinct balls into 4 distinct boxes. What is the probability that exactly two boxes are empty given that at least one box is empty?

Place 4 distinct balls into 4 distinct boxes. What is the probability that exactly two boxes are empty given that at least one box is empty?

1. There are 4 balls, and they are distinct.

2. There are 4 boxes, and they are distinct.

3. The condition event is "at least one box is empty".

4. The target event is "exactly two boxes are empty".

Find P(exactly two boxes are empty | at least one box is empty).

1. Expression

   $$A=\{\text{至少一个盒子为空}\},\; B=\{\text{恰好两个盒子为空}\}$$
   Explanation

   Set the problem's condition and target as events A and B respectively.

   Justification

   The instructor labels A and B below the problem on the screen and defines the corresponding events verbally.

   Shown in the video
2. Expression

   $$P(B\mid A)=\frac{P(AB)}{P(A)}$$
   Explanation

   Write the required conditional probability using the conditional probability formula.

   Justification

   The instructor says "according to our formula, it is P(AB) divided by $P(A)$".

   Shown in the video
3. Expression

   $$|\Omega|=4^4$$
   Explanation

   Calculate the total number of distributions.

   Justification

   Each ball has 4 choices, and multiplying for 4 balls gives $4^4$.

   Shown in the video
4. Expression

   $$P(A)=1-\frac{A_4^4}{4^4}$$
   Explanation

   Use the complementary event to find the probability of at least one empty box.

   Justification

   When there are no empty boxes, the number of ways to place 4 distinct balls into 4 distinct boxes is $A_4^4$.

   Shown in the video

The final answer is not completed within the clip; P(B|A)=$\frac{P(AB)}{P(A)}$ and $P(A)=1-\frac{A_4^4}{4^4}$ have been obtained.

Check using event inclusion: B represents exactly two empty boxes, which necessarily satisfies A representing at least one empty box, so B⊆A and P(AB)=$P(B)$. This check is added by the analyst; the clip does not continue to expand on this.

1. Caption evidence
   Observation

   Narration paraphrase: Place 4 distinct small balls into 4 distinct boxes. Find the probability that "exactly two boxes are empty" under the condition that "at least one box is empty".

2. Formula
   Observation

   Actual board temporarily displays3/32 along the complementary calculation. Editorial clarification: this is the no-empty-box probability; the source later writes1 minus3/32=$29/32$ for $P(A)$.

1. This clip only completes the establishment of the counting expression for $P(B)$, without giving the final numerical value of the conditional probability.

Place 4 distinct small balls into 4 distinct boxes. Find the probability that "exactly two boxes are empty" under the condition that "at least one box is empty".

1. The 4 balls are distinct from each other.

2. The 4 boxes are distinct from each other.

3. Each ball must be placed into some box.

4. Condition event A: At least one box is empty.

5. Target event B: Exactly two boxes are empty.

Find P(B|A).

1. Expression

   $$P(B|A)=\frac{P(AB)}{P(A)}$$
   Explanation

   First use the conditional probability formula to break down the problem.

   Justification

   Definition of conditional probability.

   Shown in the video
2. Expression

   $$P(A)=1-\frac{4!}{4^4}=\frac{29}{32}$$
   Explanation

   Editorial correction:3/32 is the complementary probability; the conditioning probability is29/32, as the source subsequently confirms.

   Justification

   The complement event is all 4 boxes being non-empty, i.e., one ball per box, totaling $A_4^4$ ways.

   Supplementary explanation
3. Expression

   $$P(AB)=P(B)$$
   Explanation

   Because "exactly two empty boxes" is necessarily included in "at least one empty box".

   Justification

   B⊆A.

   Shown in the video
4. Expression

   $$P(B)=\frac{C_4^2(C_4^3C_2^1+C_4^2)}{4^4}\ \text{（片段内未写完）}$$
   Explanation

   Start counting the arrangements for "exactly two boxes are empty": first select 2 empty boxes, then put 4 balls into the remaining 2 boxes such that neither is empty, divided into $3+1$ and $2+2$ types.

   Justification

   Classical probability counting; the video has not finished speaking about the $2+2$ type.

   Shown in the video

Editorial clarification: $P(A)=29/32$. The calculation of $P(B)$ continues in the next source segment; the intermediate3/32 denotes no empty box.

Can be verified by continuing to complete the $2+2$ type counting and substituting back into P(B|A)=$P(B)/P(A)$; this clip does not show the concluding process.

1. Caption evidence
   Observation

   Narration paraphrase: Place 4 distinct small balls into 4 distinct boxes. Find the probability that exactly two boxes are empty, given that at least one box is empty.

2. Formula
   Observation

   The screen completely boards P(B|A)=P(AB)/P(A), P(AB)=$P(B)$=..., $P(A)$=..., and the final $21/58$.

3. Audio
   Observation

   Narration paraphrase: Place 4 distinct small balls into 4 distinct boxes. Find the probability that exactly two boxes are empty, given that at least one box is empty.

1. Reviewed the original frame to confirm the combination factor $C_2^1$; the model's misreading and self-derived incorrect calculations have been corrected by this site.

Place 4 distinct small balls into 4 distinct boxes. Find the probability that exactly two boxes are empty, given that at least one box is empty.

1. The 4 balls are distinct.

2. The 4 boxes are distinct.

3. A = at least one box is empty.

4. B = exactly two boxes are empty.

Find P(B|A).

1. Expression

   $$P(B\mid A)=\frac{P(AB)}{P(A)}$$
   Explanation

   Use the conditional probability formula.

   Justification

   Definition.

   Shown in the video
2. Expression

   $$P(AB)=P(B)=\frac{C_4^2(C_4^3C_2^1+C_4^2)}{4^4}$$
   Explanation

   Since B⊆A, the probability of the intersection event equals $P(B)$; then use the counting formula to find $P(B)$.

   Justification

   Event subset relationship and classical probability.

   Shown in the video
3. Expression

   $$P(A)=1-\frac{A_4^4}{4^4}$$
   Explanation

   Use the complementary event to find the probability of at least one empty box.

   Justification

   The all-non-empty scenario is one ball per box, totaling 4! ways.

   Shown in the video
4. Expression

   $$\frac{\frac{C_4^2(C_4^3C_2^1+C_4^2)}{4^4}}{1-\frac{A_4^4}{4^4}}$$
   Explanation

   Substituting yields the fraction to be simplified.

   Justification

   Algebraic substitution.

   Derived from the video
5. Expression

   $$\frac{21}{58}$$
   Explanation

   The final result written on the video's blackboard.

   Justification

   The answer given in the video.

   Shown in the video

The video gives the answer as $21/58$.

Verified against the original frame at 266 seconds showing $C_2^1$ and 4 times 2, there are 84 ways for exactly two empty boxes and 232 ways for at least one empty box. $84/232=21/58$, consistent with the video's conclusion.

1. Caption evidence
   Observation

   Narration paraphrase: Products with 16 genuine items and 4 defective items, each with different serial numbers, are inspected. 2 items are drawn sequentially without replacement. What is the probability that the second item is also defective, given that the first item drawn is defective?

2. Caption evidence
   Observation

   Narration paraphrase: Products with 16 genuine items and 4 defective items, each with different serial numbers, are inspected. 2 items are drawn sequentially without replacement. What is the probability that the second item is also defective, given that the first item drawn is defective?

3. Audio
   Observation

   Narration paraphrase: Products with 16 genuine items and 4 defective items, each with different serial numbers, are inspected. 2 items are drawn sequentially without replacement. What is the probability that the second item is also defective, given that the first item drawn is defective?

1. The clip does not show the solution process for the second problem, only the problem and options.

Products with 16 genuine items and 4 defective items, each with different serial numbers, are inspected. 2 items are drawn sequentially without replacement. What is the probability that the second item is also defective, given that the first item drawn is defective?

1. 16 genuine items.

2. 4 defective items.

3. Total of 20 items.

4. Draw 2 items sequentially without replacement.

5. Conditioning event: First draw is defective.

6. Target event: Second draw is also defective.

Select the required conditional probability from the options.

1. Expression

   $$P(D_2\mid D_1)=\frac{P(D_1D_2)}{P(D_1)}$$
   Explanation

   Let $D_1$ and $D_2$ be drawing a defective item on the first and second draws, respectively, and use the conditional probability formula.

   Justification

   Definition of conditional probability.

   Derived from the video
2. Expression

   $$P(D_1)=\frac{4}{20}$$
   Explanation

   The probability of drawing a defective item on the first draw.

   Justification

   Classical probability.

   Derived from the video
3. Expression

   $$P(D_1D_2)=\frac{4}{20}\cdot\frac{3}{19}$$
   Explanation

   After a defective first draw, the remaining 19 items contain 3 defective ones.

   Justification

   Sampling without replacement.

   Derived from the video
4. Expression

   $$P(D_2\mid D_1)=\frac{3}{19}$$
   Explanation

   Simplify to get the conditional probability.

   Justification

   Algebraic simplification.

   Derived from the video

Based on the problem statement, the answer can be deduced as $3/19$, corresponding to option C.

This result matches option C; however, this clip does not show the instructor's solution process.

1. Diagram
   Observation

   Problem text: "Testing 16 good items and 4 defective items with different numbers, drawing 2 items sequentially without replacement. Given that the first draw is defective, the probability that the second draw is also defective is ( )".

2. Formula
   Observation

   Options are A. $1/5$, B. $3/95$, C. $3/19$, D. $1/95$.

3. Audio
   Observation

   Narration paraphrase: Testing 16 good items and 4 defective items with different numbers, drawing 2 items sequentially without replacement. What is the probability that the second draw is also defective given that the first draw is defective?

Testing 16 good items and 4 defective items with different numbers, drawing 2 items sequentially without replacement. What is the probability that the second draw is also defective given that the first draw is defective?

1. 16 good items

2. 4 defective items

3. Total 20 items

4. Drawing 2 items sequentially without replacement

5. Condition: First draw is defective

Find the probability that the second draw is defective given the first draw was defective.

1. Expression

   $$P(B\mid A)=\frac{P(AB)}{P(A)}$$
   Explanation

   First express the problem as conditional probability, then transform using the formula method.

   Justification

   Definition/Formula of conditional probability.

   Shown in the video
2. Expression

   $$P(AB)=\frac{4}{20}\times\frac{3}{19}$$
   Explanation

   Calculate the joint probability of both draws being defective.

   Justification

   Step-by-step multiplication under sampling without replacement.

   Shown in the video
3. Expression

   $$P(A)=\frac{4}{20}$$
   Explanation

   Calculate the probability that the first draw is defective.

   Justification

   Initially, the 20 products include 4 defective ones.

   Shown in the video
4. Expression

   $$P(B\mid A)=\frac{3}{19}$$
   Explanation

   After substitution, cancel the common factor $4/20$ to get the result.

   Justification

   Algebraic simplification.

   Shown in the video
5. Expression

   $$P(B\mid A)=\frac{3}{19}$$
   Explanation

   Alternatively, treat the first draw being defective as known. Among the remaining 19 items, 3 are defective, so the probability that the second draw is defective is directly $3/19$.

   Justification

   Reduced sample space method.

   Shown in the video

C. $3/19$

The video obtains the same result via two routes: the formula method simplifies to $3/19$, and the reduced sample space method directly yields $3/19$.

1. Diagram
   Observation

   The screen shows an electronic whiteboard with a light yellow text box at the top displaying the problem text, and a white area below for handwritten annotations.

2. Animation
   Observation

   Blue handwriting appears sequentially: first labeling A near the problem condition, then labeling B near the target, followed by writing P(B|A)=P(AB)/P(A), and finally writing $P(A)=1-\frac{A_4^4}{4^4}$.

1. Light yellow problem text box

2. Black printed problem text

3. Blue handwritten symbols A, B

4. Blue handwritten conditional probability formula

5. Blue handwritten $P(A)$ expression

1. Around 13 seconds, the label for event A appears.

2. Around 18 seconds, the label for event B appears.

3. Between 24 and 37 seconds, P(B|A)=P(AB)/P(A) is written step-by-step.

4. Between 49 and 95 seconds, $P(A)=1-\frac{A_4^4}{4^4}$ is written step-by-step.

1. The problem text remains in the light yellow text box at the top of the screen.

2. The handwritten derivation remains in the white area below the problem.

3. The number of balls and boxes in the problem remains 4 and 4.

The visual layout separates the original problem from the symbolic derivation: the top fixedly preserves the problem statement, while the bottom gradually converts natural language conditions into event symbols, conditional probability formulas, and classical counting expressions.

1. Diagram
   Observation

   The top yellow text box contains the original problem text; the whiteboard below has handwritten formulas in blue pen, with the conditional probability formula on the left, the calculation of $P(A)$ on the right, and subsequently P(AB)=$P(B)$ and the counting expression for $P(B)$ continued on the bottom left.

1. Problem text box

2. Handwritten formula P(B|A)=P(AB)/P(A)

3. Handwritten formula $P(A)=1-A_4^4/4^4$

4. Handwritten expansion $4\times 3\times 2$ and $4\times 4\times 4\times 4$

5. Handwritten result $3/32$

6. Handwritten P(AB)=$P(B)$

7. Beginning of handwritten counting expression for $P(B)$

1. First expand $P(A)$ from symbolic form to product form on the right, then simplify to $3/32$.

2. Subsequently write P(AB)=$P(B)$ on the bottom left.

3. Finally continue writing downwards for the numerator counting of $P(B)$, first writing $C_4^2$, then $C_4^3C_2^1$ and +$C_4^2$ inside the parentheses.