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Chapter 5: Problem 21

In a group of 20 athletes, 6 have used performance-enhancing drugs that are illegal. Suppose that 2 athletes are randomly selected from this group. Let \(x\) denote the number of athletes in this sample who have used such illegal drugs. Write the probability distribution of \(x\). You may draw a tree diagram and use that to write the probability distribution. (Hint: Note that the selections are made without replacement from a small population. Hence, the probabilities of outcomes do not remain constant for each selection.)

Short Answer

Expert verified
The probability distribution of \(x\) will be P(x=0)=\(\frac{14}{20} * \frac{13}{19}\), P(x=1)=\(\frac{14}{20} * \frac{6}{19} + \frac{6}{20} * \frac{14}{19}\), P(x=2)=\(\frac{6}{20} * \frac{5}{19}\)

Step by step solution

01

Identify Variables and Scenarios

There are 20 athletes in total, out of which 6 have used illegal performance-enhancing drugs. Therefore, there are 14 athletes who are clean. Now, 2 athletes are being selected. The variable \(x\) represents the number of athletes out of these 2 who've used drugs. So, \(x\) can be 0, 1, or 2.
02

Calculate Probability for x=0

We start off by calculating the probability that none of the selected athletes have used such drugs. This probability will be \(\frac{14}{20} * \frac{13}{19}\) because for the first athlete, we have 14 options out of 20, and for the second athlete, we have 13 remaining clean athletes out of the remaining 19.
03

Calculate Probability for x=1

On calculating the probability that one of the selected athletes has used the drugs, we have 2 possible scenarios: either the first or the second athlete used the drugs. Therefore, the probability will be the sum of both these scenarios, i.e., \(\frac{14}{20} * \frac{6}{19} + \frac{6}{20} * \frac{14}{19}\). The first term represents the case where the first athlete is clean and the second one has used drugs; the second term represents the case where the first athlete has used drugs and the second one is clean.
04

Calculate Probability for x=2

Finally, we calculate the probability that both the selected athletes have used such drugs. This probability will be \(\frac{6}{20} * \frac{5}{19}\). In this case, for the first athlete, we have 6 options out of 20, and for the second athlete, we have 5 remaining athletes who used such drugs out of the remaining 19.

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