SOLUTION: The World Bank records the prevalence of HIV in countries around the world. According to their data, "Prevalence of HIV refers to the percentage of people ages 15 to 49 who are inf

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Question 1206786: The World Bank records the prevalence of HIV in countries around the world. According to their data, "Prevalence of HIV refers to the percentage of people ages 15 to 49 who are infected with HIV." In one country, the prevalence of HIV is 16.6%. Let X = the number of people you test until you find a person infected with HIV.
What is the probability that you must test 33 people to find one with HIV? (Round your answer to four decimal places.)
What is the probability that you must ask seventeen people? (Round your answer to four decimal places.)
Answer by CPhill(1959) (Show Source):
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This problem involves a geometric distribution. A geometric random variable X represents the number of trials needed to get the first success in a sequence of independent Bernoulli trials, each with probability of success p.
In this case:
* p = 0.166 (the probability of finding an infected person)
**1. Probability of testing 33 people:**
P(X = 33) = (1-p)^(33-1) * p = (1-0.166)^32 * 0.166 ≈ 0.0005
**2. Probability of testing 17 people:**
P(X = 17) = (1-p)^(17-1) * p = (1-0.166)^16 * 0.166 ≈ 0.0091
Therefore:
* The probability of testing 33 people to find one with HIV is approximately 0.0005.
* The probability of testing 17 people to find one with HIV is approximately 0.0091.