Home / AP® Exam / AP® Statistics / 4.11 Parameters for a  Binomial Distribution- FRQs

AP Statistics 4.11 Parameters for a  Binomial Distribution- FRQs - Exam Style Questions

AP Statistics -FRQs and Answers - All Topics

Question

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.

A.
(i) Suppose one song is selected at random to be played. What is the probability that the song is a rock song? Show your work.
(ii) Suppose two songs are selected at random to be played. What is the probability that both songs are rock songs? Show your work.

B. 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.
(i) Define the random variable of interest to Ms. Fey, and state how the random variable is distributed.
(ii) What is the expected value for the random variable in part B (i)? Show your work.

C. Recall that in every one-hour period, \(20\) songs will be played at random and any song can be replayed at any time.
(i) Determine the probability that \(4\) or more rock songs in a particular one-hour period will be played. Show your work.
(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.

Most-appropriate topic codes (CED):

• TOPIC 4.3: Introduction to Probability – part A(i), A(ii)
• TOPIC 4.6: Independent Events and Unions of Events – part A(ii)
• TOPIC 4.10: Introduction to the Binomial Distribution – part B(i), C(i)
• TOPIC 4.11: Parameters for a Binomial Distribution – part B(ii), C(i)
▶️ Answer/Explanation
Detailed solution

A.
(i) \(P(\text{Rock Song}) = \frac{100}{1,000} = 0.10\)
Final Answer: 0.10

(ii) Since songs can be replayed and selections are independent:
\(P(\text{Both Rock}) = (0.10) \times (0.10) = 0.01\)
Final Answer: 0.01

B.
(i) Let \(X\) represent the number of rock songs played in one hour. Since there are \(20\) independent trials (songs played), each with probability \(p = 0.10\) of being a rock song, and songs can be replayed, \(X\) has a binomial distribution with \(n = 20\) and \(p = 0.10\).
Final Answer: Binomial distribution with n=20, p=0.10

(ii) Expected value: \(E(X) = np = 20 \times 0.10 = 2\)
Final Answer: 2 rock songs

C.
(i) \(P(X \geq 4) = 1 – P(X \leq 3) = 1 – [P(X=0) + P(X=1) + P(X=2) + P(X=3)]\)
Using binomial probabilities: \(P(X \geq 4) \approx 0.133\)
Final Answer: 0.133

(ii) No, this does not provide strong evidence that the selection process was not truly random. The probability of observing \(4\) or more rock songs in one hour is \(0.133\), which is not particularly small. This event could reasonably occur by chance alone in a random selection process.
Final Answer: No

More AP Statistics FRQs Questions..
Scroll to Top