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Chapter 7: Problem 27

At a certain medical school, \(\frac{1}{7}\) of the students are from a minority group. Of those students who belong to a minority group, \(\frac{1}{3}\) are black. a. What is the probability that a student selected at random from this medical school is black? b. What is the probability that a student selected at random from this medical school is black if it is known that the student is a member of a minority group?

Short Answer

Expert verified
a. The probability that a randomly selected student from this medical school is black is \(\frac{1}{21}\). b. The probability that a randomly selected student from this medical school is black, given that the student is a member of a minority group, is \(\frac{1}{3}\).

Step by step solution

01

Understand the given fractions

We are given that \(\frac{1}{7}\) of the students are from a minority group. Within that minority group, \(\frac{1}{3}\) of the students are black.
02

Calculate the probability of being black

To find the probability of a randomly selected student being black, we multiply the probability of being in a minority group with the probability of being black within that group. This can be represented as: \(P(Black) = P(Minority) * P(Black | Minority)\) Plugging in the given values, we have: \(P(Black) = \frac{1}{7} * \frac{1}{3}\)
03

Find the product of the fractions

To find the product of the given fractions, multiply the numerators together, and the denominators together: \(P(Black) = \frac{1* 1}{7 * 3}\)
04

Simplify the fraction

Simplify the fraction to get the probability: \(P(Black) = \frac{1}{21}\) So the probability that a randomly selected student in this medical school is black is \(\frac{1}{21}\). b. What is the probability that a student selected at random from this medical school is black if it is known that the student is a member of a minority group?
05

Understand the given information

The information we need for this question has already been given. We are asked to find the probability of a student being black given that they are already a member of a minority group.
06

Identify the probability of being black in the minority group

We are given that \(\frac{1}{3}\) of the students in the minority group are black. So the probability of being black, given that the student is a member of a minority group, is simply: \(P(Black | Minority) = \frac{1}{3}\) So the probability that a randomly selected student from this medical school is black, given that the student is a member of a minority group, is \(\frac{1}{3}\).

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