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AP Statistics 2.10 The Binomial Distribution- Exam Style Questions - MCQs - New Syllabus

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Question

Super Express Mart has a “speedy checkout” lane for people who have 6 items or less. The random variable S is the number of items purchased by a person in the “speedy checkout” lane. S follows the distribution shown in the table below.

Assuming the number of items purchased are independent, what is the probability that 3 of the next 5 people in line will have 2 or fewer items?

(A) \(\binom{5}{3}(0.10)^3(0.90)^2\)

(B) \(\binom{5}{3}(0.25)^3(0.75)^2\)

(C) \(\binom{5}{3}(0.35)^3(0.65)^2\)

(D) \((0.35)^3(0.65)^2\)

(E) \(0.10+0.25\)

▶️ Answer/Explanation

First find the probability that a person has 2 or fewer items:

\(P(S\le2)=P(S=1)+P(S=2)\)
\(=0.25+0.10\)
\(=0.35\)

Let \(X\) be the number of people among the next 5 who have 2 or fewer items. Then:

\(X\sim Binomial(n=5,p=0.35)\)

The probability that exactly 3 of the 5 people have 2 or fewer items is:

\(P(X=3)=\binom{5}{3}(0.35)^3(0.65)^2\)

This matches choice (C).
✅ Answer: (C)

Question 

Julie generates a sample of 20 random integers between 0 and 9 inclusive. She records the number of 6’s in the sample. She repeats this process 99 more times, recording the number of 6’s in each sample. What kind of distribution has she simulated?

(A) The sampling distribution of the sample proportion with \(n=20\) and \(p=0.6\)
(B) The sampling distribution of the sample proportion with \(n=100\) and \(p=0.1\)
(C) The binomial distribution with \(n=20\) and \(p=0.1\)
(D) The binomial distribution with \(n=100\) and \(p=0.1\)
(E) The binomial distribution with \(n=20\) and \(p=0.6\)

▶️ Answer/Explanation

Each random integer is equally likely to be any digit from 0 through 9. Therefore, the probability of obtaining a 6 on any draw is:
\( p=\frac{1}{10}=0.1 \)
Julie records the number of 6’s in a sample of 20 independent draws. This is a count of successes from a fixed number of trials.

Therefore, the random variable follows a binomial distribution with:
\( n=20 \)
\( p=0.1 \)
Repeating the simulation many times approximates the distribution of this binomial random variable.

✅ Answer: (C)

Question

Ms. Fey is a manager at a restaurant. The playlist contains \(1,000\) songs: \(200\) country songs, \(400\) pop songs, \(100\) rock songs, and \(300\) jazz songs. Songs are selected at random from the playlist, and any song can be replayed at any time.
In every one-hour period, \(20\) songs will be played at random. Let \(X\) represent the number of rock songs played in a one-hour period. Which of the following correctly gives the distribution of \(X\) and its expected value?
(A) \(X\sim\operatorname{Binomial}(20,0.10)\), and \(E(X)=2\)
(B) \(X\sim\operatorname{Binomial}(20,0.20)\), and \(E(X)=4\)
(C) \(X\sim\operatorname{Binomial}(1,000,0.10)\), and \(E(X)=100\)
(D) \(X\sim\operatorname{Binomial}(20,0.90)\), and \(E(X)=18\)
▶️ Answer/Explanation
Each song selection can be classified as either rock or not rock. The probability of selecting a rock song is
\(p=\frac{100}{1,000}=0.10\).
There are \(20\) song selections in one hour, and songs can be replayed, so the selections are independent. Thus,
\(X\sim\operatorname{Binomial}(n=20,p=0.10)\).
The expected value of a binomial random variable is
\(E(X)=np\).
Therefore,
\(E(X)=20(0.10)=2\).
✅ Answer: (A)
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