Question

Suppose that 5 percent of men and \(0.25\) percent of women are color-blind. A colorblind person is chosen at random. What is the probability of this person being male? Assume that there is an equal number of males and females.

Short answer

The probability that the colorblind person chosen at random is male is approximately 0.95238, or 95.24%.

Step-by-step solution

Write down the probabilities given in the exercise

We know that: 1. 5% (or 0.05) of men are colorblind: P(CB|M) = 0.05 2. 0.25% (or 0.0025) of women are colorblind: P(CB|W) = 0.0025 3. There is an equal number of males and females: P(M) = P(W) = 0.5

Write the formula for Bayes' theorem

Bayes' theorem is written as: P(A|B) = \(\frac{P(B|A) * P(A)}{P(B)}\) In our case, we want to find P(M|CB): P(M|CB) = \(\frac{P(CB|M) * P(M)}{P(CB)}\)

Calculate P(CB) using the Law of Total Probability

We can find P(CB) using the Law of Total Probability: P(CB) = P(CB|M) * P(M) + P(CB|W) * P(W) Substituting the given values: P(CB) = (0.05) * (0.5) + (0.0025) * (0.5) P(CB) = 0.02625

Calculate the probability of the person being male, given they are colorblind

Now that we have P(CB), we can use Bayes' theorem to find P(M|CB): P(M|CB) = \(\frac{P(CB|M) * P(M)}{P(CB)}\) P(M|CB) = \(\frac{(0.05) * (0.5)}{0.02625}\) P(M|CB) ≈ 0.95238 So the probability that the colorblind person chosen at random is male is approximately 0.95238, or 95.24%.

Key concepts

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. To understand this concept, let's use the example of colorblindness. If we select a person at random and are told they are colorblind, knowing their gender drastically changes the probability of them being colorblind. In the textbook exercise, the probability of a randomly chosen colorblind person being male, once we know they are colorblind, is considered conditional probability.

Mathematically, the conditional probability of A given B is denoted as P(A|B), which is the probability of event A occurring given that B is true. This concept is at the heart of Bayes' theorem, which is used to reverse conditional probabilities, allowing us to update our beliefs based on new evidence.

Law of Total Probability

The Law of Total Probability allows us to calculate the overall probability of an event based on separate conditional probabilities. It combines multiple probabilities that cover every possible scenario that could result in the event. The law states that the total probability, P(A), is the sum of the probabilities of A occurring under each given condition B1, B2, ..., Bn multiplied by the probabilities of those conditions.

Specifically, for our colorblindness case: \[ P(CB) = P(CB|M) \times P(M) + P(CB|W) \times P(W) \] By knowing the probabilities of both men and women being colorblind, along with their respective probabilities in the population, we can find the overall probability of someone being colorblind. This calculation is crucial before we can apply Bayes' theorem.

Colorblindness Probability

When dealing with colorblindness probability, we must consider the different probabilities for males and females due to genetic differences. For example, in the exercise, the probability of a man being colorblind is 5 percent, whereas for women, it's only 0.25 percent. These distinct probabilities are crucial for understanding gender-related genetic conditions like colorblindness.

To further illustrate, let's apply the probabilities to an example population with an equal number of men and women. If we pick a colorblind person at random, these gender-specific probabilities affect our estimation of the person's gender. The exercise showcases how to use these probabilities to determine the likelihood that the person is male, revealing the importance of considering gender in genetic probability models.

Probability Models

Probability models are mathematical representations of random phenomena. They help us quantify the uncertainty in different outcomes. For the colorblindness example, we use a simple model that assumes equal numbers of males and females in the population and known rates of colorblindness for each gender. Using these assumptions, we create a model that can predict the probability of a randomly selected colorblind person being male.

In practice, probability models can become complex, integrating a wide range of data and reflecting more intricate relationships between events. However, for our educational purposes, sticking to clear, simple models like the one in the exercise helps to solidify an understanding of the foundational concepts—such as conditional probability and the Law of Total Probability—in the context of real-world scenarios.