Question 1

Which of the following scenarios is best modeled by a hypergeometric distribution rather than a binomial distribution?

Accepted Answer

Drawing 5 cards from a standard deck without replacement and counting the number of aces.

Question 2

Given a population of $$N = 20$$ marbles, of which $$K = 7$$ are red, if you draw $$n = 4$$ marbles without replacement, what is the probability that exactly $$k = 2 of $$them are red? Use the hypergeometric formula.

Accepted Answer

$$P(X = 2) = \frac{\binom{7}{2} \binom{13}{2}}{\binom{20}{4}}$$

Question 3

Which of the following is NOT an assumption of the hypergeometric distribution?

Accepted Answer

Each draw is independent of the previous draws.

Question 4

A company receives a shipment of 100 units, 5 of which are defective. If 20 units are tested at random without replacement, what is the probability that exactly 2 are defective? Choose the correct formula.

Accepted Answer

$$P(X = 2) = \frac{\binom{5}{2} \binom{95}{18}}{\binom{100}{20}}$$

Question 5

Which of the following best describes the difference between the binomial and hypergeometric distributions?

Accepted Answer

The binomial distribution assumes independent trials with replacement, while the hypergeometric distribution assumes dependent trials without replacement.

Question 6

A raffle has 10 winning tickets and 90 losing tickets. If you buy 5 tickets, what is the probability that you win exactly 2 prizes? (Assume tickets are drawn without replacement.)

Accepted Answer

$$P(X = 2) = \frac{\binom{10}{2} \binom{90}{3}}{\binom{100}{5}}$$

Question 7

If the probability of finding at least 2 defective units in a sample of 25 from a shipment of 100 units (with 5 defectives) is approximately 0.095, what does this imply about the effectiveness of the testing procedure?

Accepted Answer

There is less than a 10% chance of detecting at least 2 defectives, so the procedure may not be effective enough.

Question 8

Which formula correctly gives the probability of drawing $$k$$ successes in $$n$$ draws from a population of $$N$$ items with $$K$$ successes, without replacement?

Accepted Answer

$$P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}$$

Question 9

A teacher randomly selects 4 students from a class of 30, where 12 are boys and 18 are girls. What is the probability that exactly 3 boys are selected?

Accepted Answer

$$P(X = 3) = \frac{\binom{12}{3} \binom{18}{1}}{\binom{30}{4}}$$

Question 10

Which of the following statements is TRUE regarding the hypergeometric distribution?

Accepted Answer

The probability of success changes on each draw because sampling is without replacement.

Question 11

Suppose you have a bag with 8 green and 12 yellow marbles. If you draw 5 marbles without replacement, what is the probability that all 5 are green?

Accepted Answer

$$P(X = 5) = \frac{\binom{8}{5} \binom{12}{0}}{\binom{20}{5}}$$

Question 12

A shipment contains 100 units, 5 of which are defective. If a quality control manager wants the probability of finding at least 2 defectives in a sample of 25 to be at least 10%, what should they do if the current probability is 0.095?

Accepted Answer

Increase the sample size to raise the probability above 10%.

Question 13

Which of the following is a real-world example where the binomial distribution would be inappropriate, but the hypergeometric distribution would be correct?

Accepted Answer

Selecting a committee of 3 from a group of 10 men and 5 women, and counting the number of women.

Question 14

If you draw 3 marbles from a bag containing 2 red and 4 blue marbles without replacement, what is the probability that exactly 2 are red?

Accepted Answer

$$P(X = 2) = \frac{\binom{2}{2} \binom{4}{1}}{\binom{6}{3}}$$

Question 15

A deck contains 52 cards, 4 of which are aces. If you draw 5 cards without replacement, what is the probability that you get exactly 1 ace?

Accepted Answer

$$P(X = 1) = \frac{\binom{4}{1} \binom{48}{4}}{\binom{52}{5}}$$

Question 16

Which of the following best explains why the hypergeometric distribution is used for quality control sampling?

Accepted Answer

Because items are tested without replacement, so the probability of selecting a defective item changes after each draw.

Question 17

A bag contains 15 marbles: 6 are blue and 9 are green. If you randomly select 4 marbles without replacement, what is the probability that exactly 2 are blue?

Accepted Answer

$$P(X = 2) = \frac{\binom{6}{2} \binom{9}{2}}{\binom{15}{4}}$$

Question 18

Which of the following is the mean of a hypergeometric distribution with parameters $$N$$, $$K$$, and $$n$$?

Accepted Answer

$$\mu = n \frac{K}{N}$$

Hypergeometric Distribution: Concepts, Formulas, and Applications

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