Problem 55 - A First Course in Probability

Question

In a class, there are 4 first-year boys, 6 first-year 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

There must be 9 sophomore girls present in the class for sex and class to be independent when a student is selected at random.

Step 1: Understanding Independence and the Formula

In probability theory, two events are said to be independent if the occurrence of one does not affect the probability of the occurrence of the other event. Mathematically, events A and B are independent if P(A ∩ B) = P(A)P(B). In this problem, we have two events: selecting a student based on their class (first-year or sophomore) and selecting a student based on their sex (boy or girl). We will check for independence by comparing the probabilities.

Step 2: Define the Variables

Let's define the following variables: - x: number of sophomore girls - total students: (4 + 6) first-years + (6 + x) sophomores

Step 3: Calculate Probability of each Event

We can now calculate the probability for each event: 1. P(Selecting a first-year) = (4 + 6) / (total students) 2. P(Selecting a sophomore) = (6 + x) / (total students) 3. P(Selecting a boy) = (4 + 6) / (total students) 4. P(Selecting a girl) = (6 + x) / (total students)

Step 4: Check for Independence

Next, we check for independence by comparing P(Selecting a first-year ∩ boy) with P(Selecting a first-year)P(Selecting a boy) and P(Selecting a sophomore ∩ girl) with P(Selecting a sophomore)P(Selecting a girl). 1. P(Selecting a first-year ∩ boy) = 4 / (total students) 2. P(Selecting a first-year)P(Selecting a boy) = [(4 + 6) / (total students)] * [(4 + 6) / (total students)] 3. P(Selecting a sophomore ∩ girl) = x / (total students) 4. P(Selecting a sophomore)P(Selecting a girl) = [(6 + x) / (total students)] * [(6 + x) / (total students)]

Step 5: Set up Equations for Independence and Solve

Now, we set up the equation for independence: P(Selecting a first-year ∩ boy) = P(Selecting a first-year)P(Selecting a boy) 4 / (total students) = [(4 + 6) / (total students)] * [(4 + 6) / (total students)] P(Selecting a sophomore ∩ girl) = P(Selecting a sophomore)P(Selecting a girl) x / (total students) = [(6 + x) / (total students)] * [(6 + x) / (total students)] Solving these equations, we get x = 9. So, there must be 9 sophomore girls present in the class for sex and class to be independent when a student is selected at random.

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with calculating the likelihood of various outcomes. It is essential for understanding real-world phenomena where uncertainty is present, from weather forecasts to financial markets. In our context, probability helps us determine the likelihood of selecting a student by class year or sex in a random draw.

The key point here is that probability is about predicting the chance of possible outcomes in a situation where outcomes are uncertain. However, to make accurate predictions, we need to consider how different events interact with each other—specifically, whether they are independent or not. Independence in probability means that the occurrence of one event does not influence the occurrence of another. This concept is crucial to solving the problem presented because it directly affects the calculation model to be applied.

Conditional Probability

Conditional probability is a measure of the probability of an event occurring given that another event has occurred. If we look at our scenario again, imagine we want to know the probability of selecting a girl if we know the selected student is a first-year. This is where conditional probability comes into play.

The formal definition states that the conditional probability of event A, given event B, is P(A|B) = P(A ∩ B) / P(B), as long as P(B) > 0. It's important to understand that the existence of a conditional relationship may signal that events are not independent. Returning to our exercise, in the instance where class and sex would be independent, the selection of a sophomore girl in the exercise would not be affected by knowing the student's year, and vice versa. Teaching this through step-by-step examples can make the understanding more tangible for students.

Random Variable

A random variable is a numerical description of the outcome of a statistical experiment. It assigns a number to each possible outcome of a space of events. In the context of our problem, defining a random variable, such as 'x' for the number of sophomore girls, is a technique that allows us to translate a word problem into a mathematical equation that can be solved.

With random variables, we can express complex scenarios and their probabilities in a more straightforward manner. For example, the total number of students here could be represented as a random variable that encapsulates both boys and girls from first-years and sophomores, consequently simplifying the computation of probabilities. Taking our exercise, if student's understanding can be improved by introducing the idea of a random variable early on, they can learn to manage uncertainty and structure problems effectively.