Binompdf and binomcdf functions | Random variables | AP Statistics | Khan Academy

1. Formula
   Observation

   Handwritten line: "X = # of made FTs from 7 trials with $p = 0.35$".

2. Audio
   Observation

   The explanation defines X as the free-throw success count and records the common single-shot probability.

X

Binomial random variable representing the number of made free throws out of 7 trials.

Integer-valued count; in this example, possible values are 0 through 7.

1. Formula
   Observation

   Printed problem states "I have a 0.35 probability of making a free throw." Handwritten annotation adds "$p = 0.35$".

2. Audio
   Observation

   The explanation identifies p as the single-trial success probability.

p

Probability of success on one trial, here the probability of making a single free throw.

Real number between 0 and 1; in this example $p = 0.35$.

1. Formula
   Observation

   The handwritten expression becomes "binompdf(7," and a red arrow labels the 7 as "n".

2. Audio
   Observation

   The explanation labels the trial-count argument n with a red arrow.

n

Number of trials in the stated mutually independent binomial model.

Positive integer; in this example $n = 7$.

1. Formula
   Observation

   Handwritten function name "binompdf" appears, followed by an opening parenthesis and arguments being entered.

2. Audio
   Observation

   The explanation introduces binompdf and enters its first trial-count and success-probability arguments.

1. This initial interval introduces the parameters; the same complete video later enters the remaining argument and evaluates both requested probabilities.

binompdf

Graphing-calculator function for evaluating a binomial probability mass function at a specified number of successes.

Applied to a binomial model with parameters n, p, and a target success count x.

1. Formula
   Observation

   Handwritten text defines X = # of made FTs from 7 trials.

X

The number of successful free throws (made free throws) out of the total trials.

Integer values from 0 to 7.

1. Formula
   Observation

   Handwritten text shows '7 trials' with a red arrow pointing to 'n'.

n

Number of trials in the stated mutually independent binomial model.

Positive integer; here $n = 7$.

1. Formula
   Observation

   Handwritten text shows $p = 0.35$ with a red arrow pointing to 'p'.

p

The probability of success on a single trial.

Real number between 0 and 1; here $p = 0.35$.

1. Formula
   Observation

   Handwritten expression binompdf(7, 0.35, 4).

binompdf

binompdf gives the point probability mass $P(X=x)$ for an integer count. Despite its calculator name, this is a discrete probability mass, not a continuous probability density.

Takes arguments (trials, p, x value).

1. Formula
   Observation

   The whiteboard defines X = # of made FTs from 7 trials with $p = 0.35$.

X

The number of successful free throws made out of 7 attempts.

Discrete random variable taking integer values from 0 to 7.

1. Formula
   Observation

   The whiteboard shows '7 trials' and the calculator input uses 7 for trials.

n

Number of trials in the stated mutually independent binomial model.

Positive integer, specifically $n = 7$ in this example.

1. Formula
   Observation

   The whiteboard states '$p = 0.35$' and the calculator input uses 0.35 for p.

p

The probability of success on a single trial (making a free throw).

Real number between 0 and 1, specifically $p = 0.35$.

1. Formula
   Observation

   The whiteboard shows binompdf(7, 0.35, 4) ≈ 0.14.

binompdf(n, p, x)

binompdf gives the point probability mass $P(X=x)$ for an integer count. Despite its calculator name, this is a discrete probability mass, not a continuous probability density.

Returns a real number between 0 and 1.

1. Audio
   Observation

   The explanation defines X as the free-throw success count and records the common single-shot probability.

2. Formula
   Observation

   Handwritten definition: "X = # of made FTs from 7 trials with $p = 0.35$".

Model assumption: seven mutually independent Bernoulli trials, each with the same success probability 0.35. Equal single-shot success probabilities alone do not guarantee independence.

1. Model assumption: seven mutually independent Bernoulli trials, each with the same success probability 0.35. Equal single-shot success probabilities alone do not guarantee independence.

1. Audio
   Observation

   Defining the success count lets the explanation express the first question as the exact-count event.

2. Formula
   Observation

   Handwritten expression: "$P(X = 4)$".

Once X is defined as the number of made free throws, the event "making 4 out of 7 free throws" is represented by the exact-count probability $P(X = 4)$. This is a point probability, not a cumulative probability.

1. X must be defined as the number of made free throws.

2. The desired outcome is exactly 4 made free throws out of 7.

1. Binomial random variable setup for repeated free throws

1. Audio
   Observation

   The explanation introduces binompdf and enters its first trial-count and success-probability arguments.

2. Formula
   Observation

   Handwritten expression develops into "binompdf(7, 0.35" with a red arrow labeling 7 as n.

1. This initial interval introduces the parameters; the same complete video later enters the remaining argument and evaluates both requested probabilities.

binompdf gives the point probability mass $P(X=x)$ for an integer count. Despite its calculator name, this is a discrete probability mass, not a continuous probability density.

1. Use when the random variable follows a binomial model.

2. n is the number of trials.

3. p is the probability of success on each trial.

4. x is the exact number of successes whose probability is desired.

5. At this early time the command is being introduced; the same full video later enters x and computes the answer.

1. Binomial random variable setup for repeated free throws
2. Translating the word problem into $P(X = 4)$

1. Audio
   Observation

   The lesson recommends labeling the trial parameter and illustrates it with a red arrow.

2. Formula
   Observation

   A red arrow points from the handwritten 7 in binompdf(7, 0.35 to the label "n".

The red arrow identifies the trial count n in the calculator input. Clearly identifying the parameters helps explain the calculation; this is a presentation tip, a way to make the calculation easier to follow.

1. TI-84 binompdf function for exact binomial probabilities

1. Audio
   Observation

   The calculator demonstration enters the parameters for the exact-count event and obtains the displayed decimal approximation.

2. Formula
   Observation

   Handwritten text shows binompdf(7, 0.35, 4).

binompdf gives the point probability mass $P(X=x)$ for an integer count. Despite its calculator name, this is a discrete probability mass, not a continuous probability density. Its x input is the target count, not the expectation of X.

1. The experiment must follow a binomial distribution (fixed number of independent trials, two possible outcomes, constant probability of success).

2. Model assumption: seven mutually independent Bernoulli trials, each with the same success probability 0.35. Equal single-shot success probabilities alone do not guarantee independence.

1. Formula
   Observation

   The board identifies the free-throw count X, trial count 7 and common success probability 0.35.

Model assumption: seven mutually independent Bernoulli trials, each with the same success probability 0.35. Equal single-shot success probabilities alone do not guarantee independence.

1. Model assumption: seven mutually independent Bernoulli trials, each with the same success probability 0.35. Equal single-shot success probabilities alone do not guarantee independence.

1. Formula
   Observation

   Whiteboard shows $P(X = 4)$ and binompdf(7, 0.35, 4) ≈ 0.14

binompdf gives the point probability mass $P(X=x)$ for an integer count. Despite its calculator name, this is a discrete probability mass, not a continuous probability density.

1. Requires parameters n, p, and target value x

2. Model assumption: seven mutually independent Bernoulli trials, each with the same success probability 0.35. Equal single-shot success probabilities alone do not guarantee independence.

1. Binomial Distribution Setup

1. Audio
   Observation

   The explanation converts the integer threshold to an inclusive upper bound and uses the cumulative calculator function.

2. Formula
   Observation

   Whiteboard shows $P(X < 5)$ or $P(X \le 4)$ and binomcdf(7, 0.35, 4) ≈ 0.94

To find the probability of at most x successes (or fewer than $x+1$), use the binomial cumulative distribution function (binomcdf). This calculates $P(X \le x)$.

1. Requires parameters n, p, and upper bound x

2. Useful for inequalities like $X < k$ by converting to $X \le k-1$

3. For the conversion $X<k$ to $X\le k-1$, X is integer-valued and k is an integer.

4. Model assumption: seven mutually independent Bernoulli trials, each with the same success probability 0.35. Equal single-shot success probabilities alone do not guarantee independence.

1. Binomial Distribution Setup

1. Audio
   Observation

   Defining the success count lets the explanation express the first question as the exact-count event.

2. Formula
   Observation

   Written sequence: "X = # of made FTs from 7 trials with $p = 0.35$" followed by "$P(X = 4)$".

1. Expression

   $$\text{Make 4 out of 7 free throws}$$
   Explanation

   Start from the printed question asking for the probability of making 4 out of 7 free throws when each free throw has probability 0.35.

   Justification

   Printed problem statement visible throughout the clip.

   Shown in the video
2. Expression

   $$X = \text{\# of made FTs from 7 trials with } p = 0.35$$
   Explanation

   Define a random variable X to count the number of made free throws among the 7 attempts.

   Justification

   The lecture introduces the count variable X for the printed experiment.

   Shown in the video
3. Expression

   $$P(X = 4)$$
   Explanation

   Rewrite the requested event as the probability that X takes the exact value 4.

   Justification

   Direct translation from the phrase "making 4 out of 7 free throws" once X is defined as the count of made free throws.

   Shown in the video

Defining the count X translates the first word problem to $P(X=4)$; later in this full video the calculator evaluates it. This is event translation, not a proof of the general binomial formula.

1. Audio
   Observation

   The explanation identifies fewer than five successes as the integer-count event of at most four.

2. Formula
   Observation

   Whiteboard writes $P(X < 5)$ or $P(X \le 4)$

1. Explanation

   Identify the condition 'less than 5'. Since X is discrete (integer counts), the values satisfying $X < 5$ are 0, 1, 2, 3, and 4.

   Justification

   Definition of strict inequality on integers.

   Shown in the video
2. Explanation

   Rewrite the condition as 'less than or equal to 4'. The set of values {0, 1, 2, 3, 4} is equivalent to $X \le 4$.

   Justification

   Equivalence of sets for discrete variables.

   Shown in the video
3. Explanation

   Recognize that $X \le 4$ matches the input format for the cumulative distribution function binomcdf(n, p, 4).

   Justification

   Definition of binomcdf.

   Shown in the video

$P(X < 5)$ is calculated as binomcdf(7, 0.35, 4).

1. Caption evidence
   Observation

   The printed problem gives the single-shot probability 0.35 and asks for exactly four made shots out of seven.

2. Formula
   Observation

   Handwritten setup: "X = # of made FTs from 7 trials with $p = 0.35$", then "$P(X = 4)$", then "binompdf(7, 0.35" with 7 labeled n.

3. Audio
   Observation

   The explanation introduces the trial count, common success probability and exact-count event, and labels the initial calculator parameters.

1. This initial interval introduces the parameters; the same complete video later enters the remaining argument and evaluates both requested probabilities.

Given a 0.35 probability of making a free throw, find the probability of making 4 out of 7 free throws.

1. Probability of making one free throw is 0.35.

2. There are 7 free throw attempts.

3. The desired outcome is exactly 4 made free throws.

4. Model assumption: seven mutually independent Bernoulli trials, each with the same success probability 0.35. Equal single-shot success probabilities alone do not guarantee independence.

Compute $P(X = 4)$ for the binomial random variable X counting made free throws.

1. Expression

   $$X = \text{\# of made FTs from 7 trials with } p = 0.35$$
   Explanation

   Define the random variable as the number of made free throws in 7 trials.

   Justification

   The lecturer defines X as the free-throw count.

   Shown in the video
2. Expression

   $$P(X = 4)$$
   Explanation

   Translate the requested event into probability notation.

   Justification

   The phrase "making 4 out of 7 free throws" means X equals 4 under the chosen definition of X.

   Shown in the video
3. Expression

   $$\operatorname{binompdf}(7, 0.35, \ldots)$$
   Explanation

   Begin entering the calculator function for the exact binomial probability, using 7 as n and 0.35 as p.

   Justification

   The explanation enters and labels the initial calculator arguments.

   Shown in the video

This initial interval introduces the parameters; the same complete video later enters the remaining argument and evaluates both requested probabilities.

The early interval sets up the event; the later same-video calculator result verifies approximately 0.14 under the stated model.

1. Audio
   Observation

   The calculator demonstration enters the parameters for the exact-count event and obtains the displayed decimal approximation.

2. Formula
   Observation

   Handwritten text shows $P(X=4)$ = binompdf(7, 0.35, 4) ≈ 0.14.

3. Animation
   Observation

   Calculator screen shows the input binompdf(7, 0.35, 4) and the output 0.1442381992.

Given a 0.35 probability of making a free throw, what is the probability of making exactly 4 out of 7 free throws?

1. $n = 7$ trials

2. $p = 0.35$ probability of success

3. $x = 4$ successes

4. Model assumption: seven mutually independent Bernoulli trials, each with the same success probability 0.35. Equal single-shot success probabilities alone do not guarantee independence.

Find $P(X = 4)$.

1. Explanation

   Identify the parameters of the binomial distribution: number of trials $n = 7$, probability of success $p = 0.35$, and desired number of successes $x = 4$.

   Justification

   Extracted directly from the problem statement.

   Shown in the video
2. Explanation

   Use the binompdf function on a calculator with the identified parameters.

   Justification

   The binompdf function calculates the exact probability of a specific number of successes in a binomial distribution.

   Shown in the video
3. Expression

   $$binompdf(7, 0.35, 4)$$
   Explanation

   Input the parameters into the calculator function.

   Justification

   Following the syntax binompdf(n, p, x).

   Shown in the video
4. Expression

   $$0.1442381992$$
   Explanation

   The calculator displays a finite decimal approximation to the probability.

   Justification

   Result of the calculation.

   Supplementary explanation

The probability of making exactly 4 out of 7 free throws is approximately 0.14.

The result is rounded to two decimal places as shown in the handwritten notes.

1. Formula
   Observation

   Full problem statement and solution steps visible on whiteboard and calculator screen.

Given a 0.35 probability of making a free throw, find the probability of making exactly 4 out of 7, and the probability of making less than 5 out of 7.

1. $n = 7$ trials

2. $p = 0.35$ probability of success

3. Target 1: Exactly 4 successes

4. Target 2: Less than 5 successes

5. Model assumption: seven mutually independent Bernoulli trials, each with the same success probability 0.35. Equal single-shot success probabilities alone do not guarantee independence.

Calculate $P(X = 4)$ and $P(X < 5)$.

1. Explanation

   Define the random variable X as the number of made free throws.

   Justification

   Problem setup.

   Shown in the video
2. Explanation

   For $P(X = 4)$, use the exact probability function: binompdf(7, 0.35, 4).

   Justification

   Method for exact count.

   Shown in the video
3. Explanation

   Calculate result: binompdf(7, 0.35, 4) ≈ 0.1442.

   Justification

   Calculator output shown on screen.

   Shown in the video
4. Explanation

   For $P(X < 5)$, convert to cumulative form: $P(X \le 4)$.

   Justification

   Discrete variable property.

   Shown in the video
5. Explanation

   Use the cumulative function: binomcdf(7, 0.35, 4).

   Justification

   Method for 'at most' or 'less than' conditions.

   Shown in the video
6. Explanation

   Calculate result: binomcdf(7, 0.35, 4) ≈ 0.9444.

   Justification

   Calculator output shown on screen.

   Shown in the video

$P(X = 4)$ ≈ 0.14; $P(X < 5)$ ≈ 0.94

Results match the calculator display shown in the video.

1. Diagram
   Observation

   White background with black printed text showing the free-throw probability problem and a second question about making less than 5 free throws.

2. Audio
   Observation

   The introduction presents the printed free-throw questions and the calculator-based binomial lesson.

1. Printed sentence about 0.35 probability of making a free throw

2. Printed question about making 4 out of 7 free throws

3. Printed question about making less than 5 free throws

1. No mathematical writing yet; the screen presents the problem context.

1. The success probability remains 0.35.

2. The number of trials in the first question remains 7.

The visual establishes the word problem that will be modeled as a binomial random variable.

1. Animation
   Observation

   Blue handwriting appears progressively to form "X = # of made FTs from 7 trials with $p = 0.35$".

2. Audio
   Observation

   The explanation defines X as the free-throw success count and records the common single-shot probability.

1. Variable X

2. Count of made free throws

3. Number 7

4. Parameter $p = 0.35$

1. The abstract word problem becomes a named random variable with explicit parameters.

1. The trials remain 7.

2. The success probability remains 0.35.

This visual step formalizes the probabilistic model needed before applying a calculator function.

1. Animation
   Observation

   Blue handwriting forms "binompdf(" and then fills in "7, 0.35"; a red arrow labels the 7 as "n".

2. Audio
   Observation

   The explanation introduces binompdf and enters its first trial-count and success-probability arguments.

1. This initial interval introduces the parameters; the same complete video later enters the remaining argument and evaluates both requested probabilities.

1. Function name binompdf

2. First argument 7

3. Second argument 0.35

4. Red label n

1. The probability statement $P(X = 4)$ is mapped to calculator syntax.

2. The first two parameters of the binomial model are inserted explicitly.

1. The model remains binomial.

2. The target event remains exactly 4 successes, although that value is not yet entered.

The handwriting connects the event to calculator syntax and labels the trial parameter n.

1. Animation
   Observation

   A virtual TI-84 Plus CE calculator is shown on screen. The user navigates the DISTR menu, selects binompdf, inputs the values 7, 0.35, and 4, and pastes the command to get the result.

1. Virtual TI-84 Plus CE calculator interface

2. DISTR menu

3. binompdf input screen

1. Menu scrolls from normalpdf to binompdf.

2. Values are typed into the trials, p, and x value fields.

3. The final command binompdf(7, 0.35, 4) is pasted and executed.

1. The problem statement remains visible at the top of the screen.

Demonstrates the practical steps to compute a binomial probability using a graphing calculator.

1. Animation
   Observation

   Screen recording of TI-84 Plus CE emulator showing menu navigation and calculation.

1. Calculator interface

2. DISTR menu

3. Input fields for trials, p, x value

1. User navigates to DISTR menu

2. Selects binomcdf

3. Inputs 7, 0.35, 4

4. Result 0.9443924648 appears

1. Problem context remains the same throughout the demo

Demonstrates the practical application of the binomcdf function on a standard graphing calculator used in AP Statistics.

1. Audio
   Observation

   The lesson recommends labeling the trial parameter and illustrates it with a red arrow.

2. Formula
   Observation

   The handwritten 7 in binompdf(7, 0.35 is explicitly annotated with a red arrow labeled n.

A calculator command alone always explains what its arguments mean.

Identify n as the trial count, p as the common success probability and x as the target count.

1. Audio
   Observation

   The explanation distinguishes the cumulative upper-bound input from a single exact outcome.

Thinking that the x-value in binomcdf(n, p, x) refers to the probability of exactly x successes.

The x-value in binomcdf is the upper bound of the cumulative sum, representing $P(X \le x)$, not $P(X = x)$.

1. Audio
   Observation

   Defining the success count lets the explanation express the first question as the exact-count event.

2. Formula
   Observation

   Written progression from X definition to $P(X = 4)$.

The binomial random variable definition is applied to translate the phrase "making 4 out of 7 free throws" into the exact probability statement $P(X = 4)$.

1. Audio
   Observation

   The explanation introduces binompdf and enters its first trial-count and success-probability arguments.

2. Formula
   Observation

   The board moves from $P(X = 4)$ to binompdf(7, 0.35 ...

1. This initial interval introduces the parameters; the same complete video later enters the remaining argument and evaluates both requested probabilities.

The exact-count probability $P(X = 4)$ is the type of quantity evaluated by the binomial probability mass function, which the video implements through binompdf.

1. Audio
   Observation

   The lesson recommends labeling the trial parameter and illustrates it with a red arrow.

2. Formula
   Observation

   A red arrow labels the first argument 7 as n inside the binompdf expression.

Labeling calculator parameters makes the probability calculation easier to interpret.

1. Audio
   Observation

   The explanation converts the integer threshold to an inclusive upper bound and uses the cumulative calculator function.

binompdf calculates a single point probability $P(X=k)$, while binomcdf calculates a cumulative probability $P(X\le k)$ by summing point probabilities from 0 to k.

1. Formula
   Observation

   Handwritten definition of X as the number of made free throws from 7 trials with $p = 0.35$.

1. Binomial random variable setup for repeated free throws

1. Audio
   Observation

   Defining the success count lets the explanation express the first question as the exact-count event.

2. Formula
   Observation

   Handwritten $P(X = 4)$.

1. Binomial random variable setup for repeated free throws
2. Translating the word problem into $P(X = 4)$

1. Audio
   Observation

   The explanation introduces binompdf and enters its first trial-count and success-probability arguments.

2. Formula
   Observation

   Handwritten binompdf(7, 0.35 with 7 labeled n.

1. This initial interval introduces the parameters; the same complete video later enters the remaining argument and evaluates both requested probabilities.

1. TI-84 binompdf function for exact binomial probabilities
2. Label calculator parameters

1. Audio
   Observation

   The lesson recommends labeling the trial parameter and illustrates it with a red arrow.

2. Formula
   Observation

   Red arrow labeling the 7 as n.

1. Label calculator parameters

1. Audio
   Observation

   The calculator demonstration enters the parameters for the exact-count event and obtains the displayed decimal approximation.

1. Using binompdf on a TI-84 Calculator

1. Audio
   Observation

   The explanation converts the integer threshold to an inclusive upper bound and uses the cumulative calculator function.

1. Cumulative Probability using binomcdf
2. Converting Strict Inequality to Cumulative Form

1. Audio
   Observation

   The explanation converts the integer threshold to an inclusive upper bound and uses the cumulative calculator function.

1. Exact Probability using binompdf
2. Cumulative Probability using binomcdf
3. Misinterpreting binomcdf Input