AP Statistics 2.10: Binomial Distribution
Use the binomial model by checking its conditions, calculating mean and standard deviation, and interpreting exact or cumulative probabilities.
- Syllabus
- Effective Fall 2025
- Course
- AP Statistics
Covered syllabus points
2.10.A—Justify why a random variable is or is not a binomial random variable2.10.B—Calculate the mean and standard deviation for a binomial distribution2.10.C—Interpret the mean, standard deviation, and probabilities for a binomial distribution2.10.D—Estimate probabilities of binomial random variables using data from a simulation2.10.E—Calculate probabilities for a binomial distribution
2.10 The Binomial Distribution question 1
[Maximum number: 4]
Ms. Fey is a manager at a restaurant. To improve the dining experience for her customers, she
uses a digital music service to create a playlist of songs that will be played in the restaurant.
The playlist contains 1,000 songs and consists of four different types of music in the following
quantities: 200 country songs, 400 pop songs, 100 rock songs, and 300 jazz songs. The digital
music service will select songs at random from the playlist to be played in the restaurant. Any
song can be replayed at any time.
Question (a)
(a)
In every one-hour period, 20 songs will be played at random and any song can be replayed at
any time. Ms. Fey is interested in how many rock songs will be played in a typical one-hour
period.
Question (i)
(i)
Define the random variable of interest to Ms. Fey, and state how the random variable is
distributed.
Model Solution
Let X be the number of rock songs among the 20 songs played in one hour. Since songs are selected independently with replacement and p=100/1000=0.10, X has a binomial distribution with n=20 and p=0.10.
Scoring components
1. Defines the random variable as the number of rock songs in one hour.
2. Identifies a binomial distribution.
3. States n=20 and p=0.10.
Question (ii)
(ii)
What is the expected value for the random variable in part B (i)? Show your work.
Model Solution
E(X)=np=20(0.10)=2 songs.
Scoring component
4. Correctly calculates the expected value and shows supporting work. The parameters may also be stated here.
Question (b)
(b)
Recall that in every one-hour period, 20 songs will be played at random and any song can be
replayed at any time.
Question (i)
(i)
Determine the probability that 4 or more rock songs in a particular one-hour period will
be played. Show your work.
Model Solution
P(X≥4)=1−P(X≤3)=1−x=0∑3(x20)(0.10)x(0.90)20−x=1−0.867=0.133.
Scoring components
1. Gives the correct probability.
2. Shows work supporting the probability.
Question (ii)
(ii)
Suppose 4 rock songs are played during a particular one-hour period. Does this provide
strong evidence that the song selection process was not truly random? Justify your
answer without performing an inference procedure.
Model Solution
No. The probability of four or more rock songs is 0.133, which is high enough to be reasonably attributed to chance alone and is not small enough to provide strong evidence that the selection process was not truly random.
Scoring components
3. Indicates there is not a strong reason to believe the process was not random.
4. Correctly links the probability to that decision.
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