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Chapter 7: Problem 31

Data compiled by the Department of Justice on the number of people arrested in a certain year for serious crimes (murder, forcible rape, robbery, and so on) revealed that \(89 \%\) were male and \(11 \%\) were female. Of the males, \(30 \%\) were under 18 , whereas \(27 \%\) of the females arrested were under 18 . a. What is the probability that a person arrested for a serious crime in that year was under 18 ? b. If a person arrested for a serious crime in that year is known to be under 18 , what is the probability that the person is female?

Short Answer

Expert verified
The probability that a person arrested for a serious crime in that year was under 18 is approximately \(29.88\% \), and the probability that a person arrested for a serious crime in that year is known to be under 18 is female is approximately \(9.97\% \).

Step by step solution

01

Identify the events

Let's identify the relevant events for this problem: - M: A person arrested is male. - F: A person arrested is female. - U: A person arrested is under 18.
02

Calculate the probability of each gender

We are given the probability of a person arrested being male or female. We have: - \(P(M) = 0.89\) - \(P(F) = 0.11\)
03

Calculate the probability of being under 18 given the gender

We are also given the probability of being under 18 given the person's gender: - \(P(U|M) = 0.3\) - \(P(U|F) = 0.27\)
04

Calculate the overall probability of being under 18 using the law of total probability

By the law of total probability, we can find the overall probability of a person arrested being under 18, as: \[P(U) = P(U|M)P(M) + P(U|F)P(F)\] Plugging in the values we calculated earlier: \[P(U) = (0.3)(0.89) + (0.27)(0.11)\] Calculate the probability: \[P(U) \approx 0.2988\] So, the probability that a person arrested for a serious crime in that year was under 18 is approximately \(29.88\% \).
05

Calculate the probability of being female given being under 18 using Bayes' theorem

Now we will find the probability that a person arrested for a serious crime in that year is known to be under 18 is female, using Bayes' theorem: \[P(F|U) = \frac{P(U|F)P(F)}{P(U)}\] Plugging in the values we calculated earlier: \[P(F|U) = \frac{(0.27)(0.11)}{0.2988}\] Calculate the probability: \[P(F|U) \approx 0.0997\] So, the probability that a person arrested for a serious crime in that year is known to be under 18 is female is approximately \(9.97\% \).

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