Die rolling probability with independent events | Precalculus | Khan Academy

1. Formula
   Observation

   The letter P is written to denote probability.

P

Probability of an event.

Probability notation.

1. Formula
   Observation

   The phrase 'even roll on 6-sided die' is written inside parentheses after P.

even roll on 6-sided die

The event that a single roll of a fair six-sided die produces an even number.

Event description in probability notation.

1. Formula
   Observation

   P(even roll on 6-sided die) = 3/6 = 1/2

P(even roll on 6-sided die)

Probability of rolling an even number on a single roll of a fair six-sided die.

[0, 1]

1. Formula
   Observation

   P(rolling even 3 times)

P(rolling even 3 times)

Probability of rolling an even number three consecutive times with a fair six-sided die.

[0, 1]

1. Audio
   Observation

   Narration compares the total face count with the even-face count.

2. Formula
   Observation

   The expression P(even roll on 6-sided die) = 3/6 is written.

3. Diagram
   Observation

   The numbers 1 through 6 are listed vertically, and 2, 4, and 6 are circled.

For a finite sample space of equally likely outcomes, count favorable outcomes and divide by the total.

1. All outcomes must be equally likely.

1. Audio
   Observation

   The presenter introduces independent rolls before setting up the combined event.

1. This opening interval introduces independence; the full video completes the multiplication after this interval.

Events are independent if the occurrence of one event does not affect the probability of the other event occurring.

1. The three rolls are mutually independent. Fairness alone does not establish independence.

1. Audio
   Observation

   The presenter invokes independence to justify multiplying the roll probabilities.

For this three-roll model, assume mutual independence: the probabilities for intersections of any subset factor into the individual probabilities. Pairwise independence alone is insufficient for the three-event product.

1. The three rolls are mutually independent. Fairness alone does not establish independence.

1. Formula
   Observation

   P(rolling even 3 times) = P(even roll on 6-sided die) × P(even roll on 6-sided die) × P(even roll on 6-sided die)

2. Audio
   Observation

   The narration builds a product of three single-roll probabilities under the independent-roll model.

Mutual independence of A, B and C is sufficient for the three-event intersection probability to equal the product of their marginal probabilities.

1. A, B and C are mutually independent; pairwise independence alone is not sufficient.

1. Independent Events

1. Audio
   Observation

   The calculation counts six possible faces and three even faces.

2. Formula
   Observation

   The equation P(even roll on 6-sided die) = 3/6 = 1/2 is written step-by-step.

3. Diagram
   Observation

   Numbers 1-6 are listed; 2, 4, 6 are circled to show favorable outcomes.

1. Expression

   $$\{1, 2, 3, 4, 5, 6\}$$
   Explanation

   Identify all possible outcomes of rolling a standard six-sided die.

   Justification

   Definition of the sample space for a six-sided die.

   Shown in the video
2. Expression

   $$\{2, 4, 6\}$$
   Explanation

   Identify the outcomes that satisfy the condition of being an even number.

   Justification

   Condition specified in the problem statement.

   Shown in the video
3. Expression

   $$P(\text{even}) = \frac{3}{6}$$
   Explanation

   Calculate the probability by dividing the number of favorable outcomes by the total number of outcomes.

   Justification

   A fair six-sided die: each face has equal probability.

   Supplementary explanation
4. Expression

   $$\frac{3}{6} = \frac{1}{2}$$
   Explanation

   Simplify the fraction.

   Justification

   Arithmetic simplification.

   Shown in the video

For a fair six-sided die, three of the six equally likely faces are even, giving probability 1/2.

1. Formula
   Observation

   = 1/2 × 1/2 × 1/2 = 1/8

2. Audio
   Observation

   The final calculation multiplies three one-half factors to obtain one-eighth.

1. Expression

   $$P(\text{all three even})=P(\text{even})\times P(\text{even})\times P(\text{even})$$
   Explanation

   Apply the multiplication rule for independent events.

   Justification

   The three rolls are mutually independent. Fairness alone does not establish independence.

   Supplementary explanation
2. Expression

   $$=\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}$$
   Explanation

   Substitute the probability of a single even roll (1/2) into the equation.

   Justification

   Given in the problem statement.

   Shown in the video
3. Expression

   $$=\frac{1}{8}$$
   Explanation

   Calculate the final product.

   Justification

   Arithmetic calculation.

   Shown in the video

Under fair, mutually independent rolls, the probability that all three are even is 1/8.

1. Caption evidence
   Observation

   The opening task asks for three even outcomes when rolling a die with faces numbered one through six.

2. Audio
   Observation

   The opening task asks for three even outcomes when rolling a die with faces numbered one through six.

1. This 0–74-second setup interval stops before the final calculation; the continuation in the full video supplies the result.

Find the probability of rolling even numbers three times, using a six-sided die numbered from 1 to 6.

1. A fair six-sided die: each face has equal probability.

2. Target event: every one of the 3 rolls is even.

3. The three rolls are mutually independent. Fairness alone does not establish independence.

Determine the probability of the target event occurring in all three trials.

1. Expression

   $$P(\text{even}) = \frac{3}{6} = \frac{1}{2}$$
   Explanation

   Calculate the probability of rolling an even number on a single roll.

   Justification

   Break down the compound event into simpler, single-trial events.

   Shown in the video
2. Explanation

   Recognize that the three rolls are independent events.

   Justification

   The three rolls are mutually independent. Fairness alone does not establish independence.

   Supplementary explanation

The setup establishes a per-roll probability of 1/2 and an independent-roll model; the final product appears in the continuation.

Three favorable faces out of six equally likely faces gives 3/6=1/2; this verifies the single-roll part only.

1. Caption evidence
   Observation

   The displayed problem concerns an even result on every one of three successive rolls.

2. Formula
   Observation

   P(even roll on 6-sided die) = 3/6 = 1/2

3. Formula
   Observation

   P(rolling even 3 times) = P(even roll on 6-sided die) × P(even roll on 6-sided die) × P(even roll on 6-sided die) = 1/2 × 1/2 × 1/2 = 1/8

Find the probability of rolling even numbers three times, using a six-sided die numbered from 1 to 6.

1. A fair six-sided die: each face has equal probability.

2. The probability of a single even roll is 1/2.

3. The three rolls are mutually independent. Fairness alone does not establish independence.

Calculate the probability of rolling an even number three consecutive times.

1. Expression

   $$P(\text{all three even})=P(\text{even})\times P(\text{even})\times P(\text{even})$$
   Explanation

   Since the rolls are independent events, multiply the probability of a single even roll by itself three times.

   Justification

   The three rolls are mutually independent. Fairness alone does not establish independence.

   Supplementary explanation
2. Expression

   $$=\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}$$
   Explanation

   Substitute the given probability of a single even roll.

   Justification

   Given in the problem statement.

   Shown in the video
3. Expression

   $$=\frac{1}{8}$$
   Explanation

   Perform the multiplication.

   Justification

   Arithmetic calculation.

   Shown in the video

1/8

The result is consistent with the visual representation of the calculation on the whiteboard.

1. Animation
   Observation

   Numbers 1 through 6 appear sequentially in a vertical list.

2. Animation
   Observation

   Circles are drawn around the numbers 2, 4, and 6.

1. Numbers 1-6

2. Circles

1. Numbers appear one by one.

2. Specific numbers (2, 4, 6) are highlighted with circles.

1. The list remains vertical.

2. The numbers 1, 3, 5 remain uncircled.

The visual represents the total possible outcomes of a die roll and highlights the subset of outcomes that are even numbers.

1. Animation
   Observation

   The presenter uses a dashed box to highlight and copy-paste the term 'P(even roll on 6-sided die)' multiple times to build the multiplication expression.

1. Dashed box

2. Text 'P(even roll on 6-sided die)'

1. The dashed box moves to enclose the term.

2. The term is copied and pasted to form a longer equation.

1. The value of the term remains 1/2.

This visual action emphasizes that the same probability is being multiplied for each independent roll.

1. Audio
   Observation

   The presenter rejects the belief that previous rolls change the next roll under independence.

1. The video mentions this misconception briefly; it does not present a general treatment of dependent trials.

Believing that past independent events influence the probabilities of future independent events.

Under the mutually independent-roll model, knowing previous results does not change the probability of the next even result. Fairness and independence are separate assumptions.

1. Audio
   Observation

   The lesson connects the single-roll calculation to independence of repeated rolls.

1. The multiplication is developed later in the complete video, beyond this opening interval.

Independence supplies the model assumption used to multiply the three probabilities in this example.

1. Audio
   Observation

   The narration uses independence as the model assumption for the product calculation.

Mutual independence is the sufficient model assumption used for this product calculation.

1. Formula
   Observation

   P(even roll on 6-sided die) = 3/6 = 1/2 is derived.

1. Classical Probability Formula
2. Derivation of Probability for a Single Even Roll

1. Audio
   Observation

   The presenter explains the independent-roll assumption.

1. Independence

1. Caption evidence
   Observation

   The displayed task is to obtain an even number on each of three rolls.

1. Multiplication Rule for Independent Events
2. Probability of Rolling Even Numbers Three Times

1. Audio
   Observation

   The presenter invokes independence for the multiplication step.

1. Independent Events
2. Multiplication Rule for Independent Events