Question

In a certain college, \(55 \%\) of the students are women. Suppose we take a sample of two students. Use a probability tree to find the probability (a) that both chosen students are women. (b) that at least one of the two students is a woman.

Short answer

The probability that (a) both chosen students are women is 0.3025, and (b) that at least one of the two students is a woman is 0.775.

Step-by-step solution

Determining the Probabilities for Each Branch

First, set up a probability tree with two levels, each with two branches. The first branch represents choosing a woman or a man for the first student, and the second level of branches represents choosing a woman or a man for the second student. The probability of choosing a woman is 0.55 and thus the probability of choosing a man is 1 - 0.55 = 0.45.

Calculating the Probability of Both Students Being Women

To find the probability of both students being women, follow the branches that represent choosing a woman first and a woman second. Multiply the probabilities along the branches: 0.55 (first woman) * 0.55 (second woman).

Calculating the Probability of At Least One Woman

To find this probability, calculate the probability of all scenarios with at least one woman and add them together. There are three possibilities: Woman-Woman, Woman-Man, and Man-Woman. Add the probabilities of these branches together.

Finding the Probability of Woman-Man and Man-Woman

Calculate the probability of the Woman-Man branch: 0.55 (first woman) * 0.45 (second man). Calculate the probability of the Man-Woman branch: 0.45 (first man) * 0.55 (second woman).

Summing All Probabilities for At Least One Woman

Add up the probabilities for the following branches: Woman-Woman, Woman-Man, and Man-Woman. Be careful to not include the Man-Man branch as it does not satisfy the condition of having at least one woman.

Key concepts

Probability Tree

When breaking down complicated probability problems, a probability tree is an excellent visual tool that helps elucidate possible outcomes and their associated probabilities. It's like a map that guides you through all the potential scenarios. Let's take the given exercise of selecting students as an example. Imagine two branches stemming from a starting point: one representing the selection of a woman (with a probability of 0.55) and the other a man (with a probability of 0.45). This forms the first level of the tree.

As we add a second level for the second student selection, the tree expands, showing all possible combinations of the first and second choices. By multiplying the probabilities along the branches, we can determine the likelihood of specific outcomes, such as both students being women. This is not only a practical approach to visualize the problem but also an effective way to calculate combined probabilities.

Sample Probability

The concept of sample probability is about determining the likelihood of a particular outcome within a smaller, representative group from a larger population. It's like taking a snapshot of the broader group to predict what could happen. In the college student example, we focus on the probability of a random sample of two students.

By considering the sampling of two students independently, we apply sample probability to gauge the chances of each possible pair - whether it's Woman-Woman, Woman-Man, or Man-Woman. Understanding sample probabilities allows for the prediction of outcomes even when dealing with more complex populations and larger sample sizes.

Combining Probabilities

In the world of probability, combining probabilities is a foundational skill. It involves calculating the likelihood of two or more events happening together. For instance, when we're looking at both students being women or at least one being a woman, we're essentially combining the individual probabilities of picking a woman or a man each time.

It involves simple multiplication for independent events—as seen in the Woman-Woman case (0.55 for the first and 0.55 for the second) — and addition for events that are not mutually exclusive, such as calculating the likelihood of having at least one woman in the sample. Intuitively, it's like piecing together a puzzle where each piece is a different scenario and the complete picture is the total probability for the event.

Probability Theory

At its core, probability theory is a branch of mathematics dealing with the study of random events and quantifying the likelihood of these events occurring. It's the toolbox that has everything statisticians need to analyze random phenomena. From basic operations like calculating the probability of an event to more complex concepts such as random variables and distributions, probability theory is the foundation of statistical analysis.

It equips us with principles and formulas to make sense of uncertainty. Whether we're dealing with a simple coin flip or the prediction of market trends, probability theory is the framework that helps us navigate through the inherently unpredictable nature of the world.