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Chapter 3: Problem 24

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

Short Answer

Expert verified
When there are an equal number of males and females, the probability of a randomly chosen colorblind person being male is approximately 0.9615. When there are twice as many males as females, this probability increases to approximately 0.9901.

Step by step solution

01

Understanding the given data

From the given information, we know that 5% of men and 0.25% of women are colorblind. Also, we are given that there are an equal number of males and females, so let the total population be divided as follows: 50% Males (M) 50% Females (F)
02

Representing the probabilities

We need to find the probability of a randomly chosen colorblind person being male, or P(M | Colorblind). We can represent the given data as: P(Colorblind | M) = 0.05 P(Colorblind | F) = 0.0025 P(M) = 0.5 P(F) = 0.5
03

Using Bayes' theorem

To find P(M | Colorblind), we use Bayes' theorem, which states: P(M | Colorblind) = \(\frac{P(Colorblind | M) * P(M)}{P(Colorblind)}\) We also need to find P(Colorblind), which can be calculated using the law of total probability: P(Colorblind) = P(Colorblind | M) * P(M) + P(Colorblind | F) * P(F)
04

Calculating the probability

Substitute the given values in the above equations: P(Colorblind) = (0.05 * 0.5) + (0.0025 * 0.5) = 0.02625 Now, we can find the probability of the colorblind person being male: P(M | Colorblind) = \(\frac{(0.05 * 0.5)}{0.02625}\) = 0.96153846 Phase 2: Twice as many males as females
05

Changing the population distribution

Now, we are given that the population consists of twice as many males as females. Let's redefine the population distribution as follows: \( \frac{2}{3} \) Males (M) \( \frac{1}{3} \) Females (F) Also, update the probabilities using the new population distribution: P(M) = \( \frac{2}{3} \) P(F) = \( \frac{1}{3} \)
06

Calculating the probability for the new distribution

We use the same equations as before with the new values to find P(M | Colorblind): P(Colorblind) = (0.05 * \( \frac{2}{3} \)) + (0.0025 * \( \frac{1}{3} \)) = \( \frac{101}{600} \) Now, we can find the probability of the colorblind person being male with the new distribution: P(M | Colorblind) = \(\frac{(0.05 * \frac{2}{3})}{\frac{101}{600}}\) = 0.99009901. So, when there are an equal number of males and females, the probability of a randomly chosen colorblind person being male is approximately 0.9615, and when there are twice as many males as females, this probability increases to approximately 0.9901.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Calculation
Understanding probability is crucial for interpreting a variety of real-world phenomena, including genetics and health-related issues like color blindness. Probability calculation is essentially quantifying the chance that a particular event will occur. It is expressed as a value between 0 and 1, where 0 means the event will not occur, and 1 indicates certainty that the event will occur.

For example, in the given exercise, the probabilities of men and women being colorblind, denoted as P(Colorblind | M) and P(Colorblind | F), are given as 0.05 and 0.0025, respectively. We also learn that the population is evenly split between males (M) and females (F). Therefore, the probability of randomly picking a male or a female from the population is P(M) = 0.5 and P(F) = 0.5. Understanding how to interpret these probabilities and calculating the overall chances of an event, like selecting a male who is colorblind from the entire population, involves using these fundamental probability concepts and applying Bayes' theorem for conditional probabilities.
Color Blindness Probability
In the context of genetics and probability, it's interesting to look at specific conditions, such as color blindness, to understand how these probabilities affect populations. Color blindness is a genetic condition that affects a person's ability to perceive color, and it is more common in males than females due to its inheritance patterns.

The question in the exercise asks us to figure out the conditional probability of a person being male given they are colorblind. This involves understanding the conditional probability P(Colorblind | M), which is the probability of a male being colorblind, and P(Colorblind | F), the probability of a female being colorblind. These probabilities are essential for calculating the likelihood of a randomly chosen colorblind person being male, a concept which can be extended to other genetically inherited conditions.
Law of Total Probability
The law of total probability is a fundamental rule that provides a way to break down complex probability problems into simpler parts. In the case of color blindness, this law helps us determine the overall probability of someone being colorblind, regardless of their sex.

To calculate this, we use the information provided in the exercise about each subgroup, males and females, and combine these probabilities to get the total probability of color blindness in the population. Specifically, we calculate P(Colorblind) by summing the products of the probabilities of being colorblind given being male, P(Colorblind | M), and the probability of being male, P(M), with the similar product for females. The law of total probability is crucial when the condition can arise from multiple sources or categories, allowing a complex probability to be understood in a step-by-step manner.

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Most popular questions from this chapter

Consider two boxes, one containing 1 black and 1 white marble, the other 2 black and 1 white marble. A box is selected at random, and a marble is drawn at random from the selected box. What is the probability that the marble is black? What is the probability that the first box was the one selected, given that the marble is white?

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