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AP Statistics 4.10 Introduction to the Binomial Distribution- MCQs - Exam Style Questions

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Question

A wildlife scientist is investigating the percentage of baby elephants in a certain country that can stand on their hind legs to reach food. Previous studies have found that about \(20\%\) of baby elephants can do so. Let \(G\) represent the number of baby elephants who can stand on their hind legs in a random sample of 60 baby elephants from the country.
Which of the following is the best interpretation of the expression \(P(G=12)\)?
(A) The expression represents the expected number of baby elephants in a random sample of 60 who can stand on their hind legs.
(B) The expression represents the probability that 12 or fewer baby elephants from all baby elephants in the country can stand on their hind legs.
(C) The expression represents the probability that 12 or fewer baby elephants from a random sample of 60 can stand on their hind legs.
(D) The expression represents the probability that exactly 12 baby elephants from all baby elephants in the country can stand on their hind legs.
(E) The expression represents the probability that exactly 12 baby elephants from a random sample of 60 in the country can stand on their hind legs.
▶️ Answer/Explanation
Detailed solution
\(G\) counts successes in a sample of size \(n=60\).
Interpreting \(P(G=12)\): the probability the sample contains exactly 12 successes (elephants that can stand) out of the 60 chosen.
That matches the description in option (E).
✅ Answer: (E)

Question

Based on his past record, Luke, an archer for a college archery team, has a probability of 0.90 of hitting the inner ring of the target with a shot of the arrow. Assume that in one practice Luke will attempt 5 shots of the arrow and that each shot is independent from the others. Let the random variable $X$ represent the number of times he hits the inner ring of the target in 5 attempts. The probability distribution of $X$ is given in the table.
$X$012345
$P(X)$0.000010.000450.008100.072900.328050.59049
What is the probability that the number of times Luke will hit the inner ring of the target out of the 5 attempts is less than the mean of $X$?
(A) 0.40951
(B) 0.50000
(C) 0.59049
(D) 0.91854
(E) 0.99144
▶️ Answer/Explanation
Detailed solution

1. Calculate the Mean of X:
This is a binomial distribution with $n=5$ attempts and probability of success $p=0.90$.
The mean (expected value) is $E(X) = n \times p$.
$E(X) = 5 \times 0.90 = 4.5$

2. Determine the Required Probability:
We need to find the probability that the number of hits is *less than the mean*. So we need to find $P(X < 4.5)$.
Since $X$ must be an integer, this is the same as $P(X \le 4)$.

3. Calculate the Probability:
Using the complement rule is fastest:
$P(X \le 4) = 1 – P(X > 4) = 1 – P(X=5)$.
From the table, $P(X=5) = 0.59049$.

$P(X < 4.5) = 1 – 0.59049 = 0.40951$
✅ Answer: (A)

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