Problem 26 - A First Course in Probability

Question

Suppose that 5 percent of men and .25 percent of women are color blind. A color-blind person is chosen at random. What is the probability of this person being male? Assume that there are an equal number of males and females. What if the population consisted of twice as many males as females?

Step-by-Step Solution

Answer

When there are an equal number of males and females, the probability of the color-blind person being male is approximately \(0.952381\). If there are twice as many males as females, the probability increases to approximately \(0.975823\).

Step 1: Identify the Given Probabilities

We are given that 5 percent of men are color blind, and .25 percent of women are color blind. These values can be expressed as: P(Color blind | Male) = 0.05 P(Color blind | Female) = 0.0025 Step 2: Calculate the Prior Probabilities

Step 2: Calculate the Prior Probabilities

In this case, we have an equal number of males and females. This means that the prior probabilities are: P(Male) = 0.5 P(Female) = 0.5 Step 3: Calculate the Joint Probabilities

Step 3: Calculate the Joint Probabilities

Now calculate the joint probabilities, by multiplying the conditional probabilities by the respective prior probabilities: P(Color blind, Male) = P(Color blind | Male) * P(Male) = 0.05 * 0.5 = 0.025 P(Color blind, Female) = P(Color blind | Female) * P(Female) = 0.0025 * 0.5 = 0.00125 Step 4: Calculate the Marginal Probability

Step 4: Calculate the Marginal Probability

Calculate the marginal probability of a color-blind person: P(Color blind) = P(Color blind, Male) + P(Color blind, Female) = 0.025 + 0.00125 = 0.02625 Step 5: Calculate the Probability of Male Given Color Blindness

Step 5: Calculate the Probability of Male Given Color Blindness

Finally, using Bayes' theorem, find the probability of a color-blind person being male: P(Male | Color blind) = P(Color blind, Male) / P(Color blind) = 0.025 / 0.02625 ≈ 0.952381 For the case where there are twice as many males as females: Step 6: Update the Prior Probabilities

Step 6: Update the Prior Probabilities

When there are twice as many males as females in the population, the prior probabilities are: P(Male) = 1/3 P(Female) = 2/3 Step 7: Recalculate the Joint Probabilities

Step 7: Recalculate the Joint Probabilities

Recalculate the joint probabilities: P(Color blind, Male) = 0.05 * 2/3 ≈ 0.03333 P(Color blind, Female) = 0.0025 * 1/3 ≈ 0.00083 Step 8: Recalculate the Marginal Probability

Step 8: Recalculate the Marginal Probability

Recalculate the marginal probability of a color-blind person: P(Color blind) = 0.03333 + 0.00083 = 0.03416 Step 9: Recalculate the Probability of Male Given Color Blindness

Step 9: Recalculate the Probability of Male Given Color Blindness

Recalculate the probability using Bayes' theorem: P(Male | Color blind) = 0.03333 / 0.03416 ≈ 0.975823 So when there are an equal number of males and females, the probability of the color-blind person being male is approximately 0.952381. If there are twice as many males as females, the probability increases to approximately 0.975823.

Key Concepts

Conditional Probability

Conditional probability helps us calculate the likelihood of an event happening when we know that another event has already occurred.
In this problem, we are given that a certain percentage of men and women are color blind. These are expressed as conditional probabilities:

• \( P(\text{Color blind} \mid \text{Male}) = 0.05 \)
• \( P(\text{Color blind} \mid \text{Female}) = 0.0025 \)

These equations tell us the probability of someone being color blind if we know their gender. Conditional probability plays a crucial role in understanding how different factors can affect outcomes. It's a foundational concept in probability theory that helps us refine our predictions in the presence of additional information.

Joint Probability

Joint probability is all about finding the probability of two or more events happening at the same time. In our exercise, we look at the probability of being both color blind and male or female. This is calculated using the conditional probabilities and the prior probabilities of being male or female.
The formula for joint probability is:

• \( P(A \text{ and } B) = P(A \mid B) \cdot P(B) \)

For example, in our exercise:

• \( P(\text{Color blind, Male}) = P(\text{Color blind} \mid \text{Male}) \times P(\text{Male}) \)
• Resulting in \( P(\text{Color blind, Male}) = 0.05 \times 0.5 = 0.025 \)

Understanding joint probabilities is essential for calculating the likelihood of overlapping events and building foundational skills in probability.

Marginal Probability

Marginal probability represents the probability of an event occurring, regardless of whether another event happens. It's like taking a step back and looking at a broader picture.
In our problem, we calculate the marginal probability of a person being color blind:

• \( P(\text{Color blind}) = P(\text{Color blind, Male}) + P(\text{Color blind, Female}) \)

This means adding up the probabilities of all scenarios where the event in question (being color blind) occurs.
In the context of this exercise, it helps establish a base for calculating conditional probabilities using Bayes' theorem. Marginal probabilities give us an overarching view of an event's likelihood.

Bayes' Theorem

Bayes' Theorem provides a way to update our beliefs in the presence of new data. It links together conditional and marginal probabilities to give us a more complete picture.
The theorem is expressed as:

• \( P(B \mid A) = \frac{P(A \mid B) \times P(B)}{P(A)} \)

By using Bayes' Theorem, we assess the probability of one event given the occurrence of another. In our exercise, we use it to calculate the probability of being male if a person is color blind:

• \( P(\text{Male} \mid \text{Color blind}) = \frac{P(\text{Color blind, Male})}{P(\text{Color blind})} \)

In both equal and unequal male-female populations, we see how Bayes' Theorem provides clarity by integrating known probabilities into a cohesive and updated assessment. It's a powerful tool for decision-making and risk assessment based on evolving information.