Problem 55 - A First Course in Probability

Question

In a class, there are 4 freshman boys, 6 freshman girls, and 6 sophomore boys. How many sophomore girls must be present if sex and class are to be independent when a student is selected at random?

Step-by-Step Solution

Answer

To ensure that sex and class are independent when a student is selected at random, there must be 9 sophomore girls present in the class.

Step 1: Understand Independence

Two events A and B are independent if the probability of occurrence of one event does not depend on the occurrence of the other event. Mathematically, it can be written as: \(P(A \cap B) = P(A) \times P(B)\) In terms of our specific problem - A represents the event of selecting a student based on class (freshman or sophomore), and B represents the event of selecting a student based on sex (boy or girl). We will use this concept to find the necessary number of sophomore girls.

Step 2: Calculate the Probabilities of Each Event

Given: - 4 freshman boys - 6 freshman girls - 6 sophomore boys Let x be the number of sophomore girls. The total number of students = 4 + 6 + 6 + x. The probability of selecting a freshman: \(P(\text{Freshman}) = \frac{4 + 6}{4 + 6 + 6 + x} = \frac{10}{16 + x}\) The probability of selecting a boy: \(P(\text{Boy}) = \frac{4 + 6}{4 + 6 + 6 + x} = \frac{10}{16 + x}\) The probability of selecting a freshman boy: \(P(\text{Freshman} \cap \text{Boy}) = \frac{4}{4 + 6 + 6 + x}\)

Step 3: Apply the Independence Formula

According to the independence rule, the probability of selecting a freshman boy can be calculated as: \(P(\text{Freshman} \cap \text{Boy}) = P(\text{Freshman}) \times P(\text{Boy})\) Substituting the probabilities: \(\frac{4}{16 + x} = \frac{10}{16 + x} \times \frac{10}{16 + x}\)

Step 4: Solve for Required Number of Sophomore Girls

Now, we need to solve the equation to find the value of x (the number of sophomore girls needed to ensure independence): \(\frac{4}{16 + x} = \frac{10 \times 10}{(16 + x)^2}\) Simplifying and solving for x: \(4(16 + x) = 100\) \(64 + 4x = 100\) \(4x = 36\) \(x = 9\) So, to ensure that sex and class are independent when a student is selected at random, there must be 9 sophomore girls present in the class.

Key Concepts

Conditional Probability

Understanding conditional probability is essential in determining how the occurrence of one event affects the probability of another. In simple terms, it's the probability that event B happens given that event A has already occurred, and is denoted by the formula:

P(B|A) = \( \frac{P(A \cap B)}{P(A)} \)

This formula is pivotal in complex probability questions where events are interrelated. To apply this to our classroom example, if we wanted to know the probability of selecting a sophomore given the student is a girl, we'd use this concept. In educational scenarios, conditional probability helps provide clarity on how two criteria, such as 'sex' and 'class year,' interact with each other within a given population.

Probability Formula

The general probability formula is the backbone of solving most probability exercises. It is expressed as:

P(Event) = \( \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}} \)

This fundamental formula guides our approach to solving problems by clearly defining our 'favourable outcomes' and 'total outcomes.' For the problem at hand, we calculated the probabilities of being a freshman, being a boy, and being both a freshman and a boy, to solve for the number of sophomore girls required for sex and class to be independent. A student's grasp of this formula is crucial as it is the stepping stone to tackling more complex problems in probability.

Independent Events

Independent events are a key concept where the occurrence of one event does not influence the likelihood of the other. In mathematical terms, two events A and B are independent if:
\(P(A \cap B) = P(A) \times P(B)\)

In our classroom situation, 'being a freshman' and 'being a boy' are two separate events that would be independent if the probability of their intersection equals the product of their individual probabilities. When we solve for the number of sophomore girls needed, it enables the calculation of complete independence between these two factors, sex and class. Students who master this concept can analyze more complex scenarios involving multiple events with confidence and precision.