Problem 20 - A First Course in Probability

Question

Fifty-two percent of the students at a certain college are females. Five percent of the students in this college are majoring in computer science. Two percent of the students are women majoring in computer science. If a student is selected at random, find the conditional probability that (a) the student is female given that the student is majoring in computer science; (b) this student is majoring in computer science given that the student is female.

Step-by-Step Solution

Answer

In conclusion: (a) The conditional probability that a student is female given that the student is majoring in computer science is \(40\%\). (b) The conditional probability that a student is majoring in computer science given that the student is female is \(3.85\%\).

Step 1: Understand the conditional probability formula

The formula for conditional probability is: P(A | B) = P(A ∩ B) / P(B). For our problems, we need to find: (a) P(Female | Computer Science) (b) P(Computer Science | Female) Step 2: Calculate the probability of Female and Computer Science

Step 2: Calculate the intersection of Female and Computer Science

We are given that 2% of the students are women majoring in computer science, which is the same as the probability of the intersection of Female and Computer Science. So, P(Female ∩ Computer Science) = 0.02. Step 3: Calculate the conditional probability of a student being female given that the student is majoring in computer science

Step 3: Calculate P(Female | Computer Science)

To find P(Female | Computer Science), we use the conditional probability formula: P(Female | Computer Science) = P(Female ∩ Computer Science) / P(Computer Science) We have P(Female ∩ Computer Science) = 0.02 and P(Computer Science) = 0.05. So, P(Female | Computer Science) = 0.02 / 0.05 = 0.4 = 40%. Step 4: Calculate the conditional probability of a student majoring in computer science given that the student is female

Step 4: Calculate P(Computer Science | Female)

To find P(Computer Science | Female), we use the conditional probability formula: P(Computer Science | Female) = P(Female ∩ Computer Science) / P(Female) We have P(Female ∩ Computer Science) = 0.02 and P(Female) = 0.52. So, P(Computer Science | Female) = 0.02 / 0.52 ≈ 0.0385 ≈ 3.85%. In conclusion: (a) The conditional probability that a student is female given that the student is majoring in computer science is \(40\%\). (b) The conditional probability that a student is majoring in computer science given that the student is female is \(3.85\%\).

Key Concepts

Probability Theory

Probability theory is a fundamental branch of mathematics, focusing on the study of random events and the likelihood of different outcomes. It allows us to make predictions about uncertain situations. The basic idea is to quantify how likely events are to occur, using numbers within a range from 0 to 1.

• A probability of 0 means the event is impossible.
• A probability of 1 indicates certainty.
• Probabilities between 0 and 1 denote the varying degrees of likelihood.

One important concept within probability theory is the idea of a sample space, the set of all possible outcomes for an event. For example, when flipping a coin, the sample space is heads or tails. Understanding how to calculate probabilities helps in everyday decisions, such as assessing risks or understanding trends in data.

Intersection of Events

In probability theory, the intersection of events is a critical concept used to understand situations where multiple conditions are met simultaneously. The intersection of events A and B, denoted as \( A \cap B \), refers to the outcome set where both events occur.
Suppose you have two events: Event A, where a student is female, and Event B, where a student majors in computer science. The probability of the intersection, \( P(A \cap B) \), is 0.02, meaning that 2% of students are women majoring in computer science.

• The intersection helps us focus on the probability of combined characteristics.
• In many real-life applications, knowing the intersection is vital for understanding frequency and relationship between events.

Bayes' Theorem

Bayes' Theorem is a powerful tool in the field of probability theory that provides a way to update our knowledge based on new information. It is often used to calculate conditional probabilities, where we are interested in the probability of an event given that another event has occurred.
Bayes' Theorem can be expressed with the formula:\[P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)}\]This formula might seem complex at first glance, but it helps solve problems where direct computation might not be feasible. Although not directly used in our initial exercise, understanding Bayes' Theorem provides valuable insights into more advanced topics like statistical inference, decision making, and machine learning.

Probability Calculations

Probability calculations involve several steps and methods to determine the likelihood of different events or scenarios. To calculate conditional probabilities, for example, you divide the probability of the intersection of two events by the probability of the given event.
In the provided exercise:

• To find \( P(\text{Female} | \text{Computer Science}) \), divide \( P(\text{Female} \cap \text{Computer Science}) \) by \( P(\text{Computer Science}) \). This gives \( 0.02 / 0.05 = 0.4 \), or 40%.
• For \( P(\text{Computer Science} | \text{Female}) \), divide \( P(\text{Female} \cap \text{Computer Science}) \) by \( P(\text{Female}) \). This results in \( 0.02 / 0.52 \approx 0.0385 \), or 3.85%.

These calculations help us understand relationships between different characteristics, allowing for more informed predictions and decisions.