Problem 27 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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?

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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: \(P(Black) = \frac{1}{7} * \frac{1}{3}\)

Step 3: Find the product of the fractions

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

Step 4: 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?

Step 1: 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.

Step 2: 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 \(\frac{1}{3}\).

Key Concepts

Probability Calculations

Understanding probability calculations is fundamental in mathematics. Probability represents the chance that a particular event will occur. It is calculated as the ratio of the number of favorable outcomes to the number of all possible outcomes. When we have a situation where events are dependent on each other, like picking a black student from a minority group, we utilize the basic principles of probabilities.

For instance, in our medical school scenario, to find the overall probability of selecting a black student, we need to consider all students, not just those within the minority group. The choice is structured as an 'and' situation where a student must be both in the minority group and be black, which involves multiplying the chances for each independent probability. Thus, the calculation would look something like \( P(Black) = P(Minority) * P(Black | Minority) \), where \( P(Black | Minority) \) is the conditional probability of a student being black given that they belong to a minority.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. In symbols, this is expressed as \( P(A|B) \), where the vertical bar '|' represents 'given'. This measures the probability of event A occurring if it's known that event B has already taken place.

In the problem from the medical school, part b asks us to find the probability that a student is black knowing that they are in a minority group. Here, the probability of being black is conditioned on the student already being part of the minority. Thus, the calculation is straightforward, as we only consider the subset of minority students. This awareness of the subset changes the scope of our probability calculation, which is the essence of conditional probability.

Fraction Multiplication

When multiplying fractions, we follow a simple rule: multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. This rule is used when calculating probabilities involving multiple independent events, such as finding the probability of randomly selecting a black student from the minority student population at the medical school.

We express these probabilities as fractions; for the given problem, the probability of picking a black student from the minority group is calculated by multiplying \(\frac{1}{7}\) by \(\frac{1}{3}\). The multiplication results in \(P(Black) = \frac{1* 1}{7 * 3}\), which simplifies to \(P(Black) = \frac{1}{21}\). It's important to simplify these fractions to their lowest terms to accurately express the probability.