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AP Statistics 4.4 Mutually Exclusive Events- FRQs - Exam Style Questions

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Question

Nine sales representatives, \(6\) men and \(3\) women, at a small company wanted to attend a national convention. There were only enough travel funds to send \(3\) people. The manager selected \(3\) people to attend and stated that the people were selected at random. The \(3\) people selected were women. There were concerns that no men were selected to attend the convention.

(a) Calculate the probability that randomly selecting \(3\) people from a group of \(6\) men and \(3\) women will result in selecting \(3\) women.

(b) Based on your answer to part (a), is there reason to doubt the manager’s claim that the \(3\) people were selected at random? Explain.

(c) An alternative to calculating the exact probability is to conduct a simulation to estimate the probability. A proposed simulation process is described below.

Each trial in the simulation consists of rolling three fair, six-sided dice, one die for each of the convention attendees. For each die, rolling a \(1\), \(2\), \(3\), or \(4\) represents selecting a man; rolling a \(5\) or \(6\) represents selecting a woman. After \(1,000\) trials, the number of times the dice indicate selecting \(3\) women is recorded.

Does the proposed process correctly simulate the random selection of \(3\) women from a group of \(9\) people consisting of \(6\) men and \(3\) women? Explain why or why not.

Most-appropriate topic codes (CED):

• TOPIC 4.4: Mutually Exclusive Events — part (a)
• TOPIC 4.3: Introduction to Probability — part (b)
• TOPIC 4.2: Estimating Probabilities Using Simulation — part (c)
▶️ Answer/Explanation
Detailed solution

(a)
\(P(\text{3 women}) = \frac{3}{9} \times \frac{2}{8} \times \frac{1}{7} = \frac{6}{504} = \frac{1}{84} \approx 0.0119\)
Answer: \(\boxed{0.012}\)

(b)
Yes, there is reason to doubt the manager’s claim. The probability of selecting \(3\) women by random chance is only about \(1.2\%\), which is very small. This suggests it’s unlikely this outcome occurred purely by chance.

(c)
No, the proposed process does not correctly simulate the random selection. The dice simulation assumes:
• Independent selections (dice rolls are independent)
• Constant probability of selecting a woman (\( \frac{2}{6} = \frac{1}{3} \))

However, the actual selection is:
• Dependent selections (sampling without replacement)
• Changing probabilities (after each woman is selected, probability decreases)

The dice simulation represents sampling with replacement, not the actual without-replacement scenario.

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