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Chapter 3: Problem 26

Suppose that 5 percent of men and 0.25 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?

Short Answer

Expert verified
The probability of a randomly chosen color-blind person being male, given an equal number of males and females, is approximately 0.9518 or 95.18%. When there are twice as many males as females, the probability increases to approximately 0.9804 or 98.04%.

Step by step solution

01

Equal Number of Males and Females

First let's solve the problem with an equal number of males and females. We can use the conditional probability formula: \[P(A \mid B) = \frac{P(A \cap B)}{P(B)}\] where A is the event of being male and B is the event of being color-blind. We need to find \(P(\text{Male} \mid \text{Color-blind})\). Let M be the event of the person being male, and C be the event of the person being color-blind. So, we have: \[P(\text{Male} \mid \text{Color-blind}) = \frac{P(\text{Male} \cap \text{Color-blind})}{P(\text{Color-blind})}\] Now, we are given: \[P(\text{Color-blind} \mid \text{Male}) = 0.05\] and \[P(\text{Color-blind} \mid \text{Female}) = 0.0025\] Since there are equal numbers of males and females, the probability of any person being male or female can be assumed as \(P(\text{Male}) = P(\text{Female}) = 0.5\). Using the law of total probability, we can find \(P(\text{Color-blind})\): \[P(\text{Color-blind}) = P(\text{Color-blind} \mid \text{Male}) \times P(\text{Male}) + P(\text{Color-blind} \mid \text{Female}) \times P(\text{Female})\] \[P(\text{Color-blind}) = (0.05 \times 0.5) + (0.0025 \times 0.5) = 0.02625\] Now we can find the probability of a color-blind person being male using the conditional probability formula: \[P(\text{Male} \mid \text{Color-blind}) = \frac{P(\text{Male} \cap \text{Color-blind})}{P(\text{Color-blind})}\] \[P(\text{Male} \mid \text{Color-blind}) = \frac{0.05 \times 0.5}{0.02625} 鈮� 0.9518\] So, the probability of a randomly chosen color-blind person being male, given an equal number of males and females, is approximately 0.9518 or 95.18%.
02

Twice as Many Males as Females

Now let's solve the problem with twice as many males as females. In this case, the probability of any person being male or female can be assumed as \(P(\text{Male}) = \frac{2}{3}\) and \(P(\text{Female}) = \frac{1}{3}\). Using the law of total probability, we can find \(P(\text{Color-blind})\): \[ P(\text{Color-blind}) =P(\text{Color-blind} \mid \text{Male}) \times P(\text{Male}) + P(\text{Color-blind} \mid \text{Female}) \times P(\text{Female}) \] \[P(\text{Color-blind}) = (0.05 \times \frac{2}{3}) + (0.0025 \times \frac{1}{3}) = 0.034167\] Now we can find the probability of a color-blind person being male using the conditional probability formula: \[P(\text{Male} \mid \text{Color-blind}) = \frac{P(\text{Male} \cap \text{Color-blind})}{P(\text{Color-blind})}\] \[P(\text{Male} \mid \text{Color-blind}) = \frac{0.05 \times \frac{2}{3}}{0.034167} 鈮� 0.9804\] Thus, the probability of a randomly chosen color-blind person being male when there are twice as many males as females is approximately 0.9804 or 98.04%.

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Key Concepts

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

Probability Theory
Probability theory is the branch of mathematics that deals with the analysis of random events and the likelihood of their occurrence. Essentially, it's about quantifying how likely it is for a particular event to happen. For instance, if we're flipping a fair coin, probability theory tells us that there's a 50% chance of landing on heads and a 50% chance of tails.

It involves understanding and calculating probabilities, which are expressed as a number between 0 and 1. A probability of 0 means the event will never occur, while a probability of 1 means it's certain to happen. Events with probabilities closer to 1 are more likely to occur than events with probabilities closer to 0. In the context of the textbook exercise on color-blindness, probability theory is applied to determine the likelihood of a color-blind person being male.
Law of Total Probability
The law of total probability is a fundamental rule that relates marginal probabilities to conditional probabilities. It helps us to calculate the probability of a general event by considering all possible ways that event can occur. The law states that if you have a set of disjoint events that together cover the entire sample space, the probability of another event can be calculated as the sum of its conditional probabilities on each of the disjoint events.

In mathematical terms, if we have mutually exclusive events B1, B2,..., Bn that cover the whole sample space, then:
\[P(A) = P(A \mid B_1)P(B_1) + P(A \mid B_2)P(B_2) + ... + P(A \mid B_n)P(B_n)\]
Using the law of total probability, we break down complex probability questions into smaller parts, which makes the overall calculation more manageable.
Color Blindness Probability
The probability of color blindness among a population demonstrates how probability theory can be used to understand genetic conditions. Color blindness is a condition where individuals have difficulty distinguishing between certain colors, most commonly red and green. The probability of someone being color-blind can differ between populations and genders, as highlighted in the exercise.

To calculate these probabilities, we can apply knowledge of how traits like color blindness are inherited and use statistical data. Since men are generally more likely to be color-blind (due to the X-linked nature of the most common forms of color blindness), gender plays a crucial role in calculating this type of probability.
Bayes' Theorem Applications
Bayes' theorem is a powerful tool in probability theory for finding conditional probabilities. It is frequently used in various fields, including statistics, finance, medicine, and even machine learning. The theorem provides a way to update the probability estimate for an event based on new evidence.

The formula for Bayes' theorem is given by:
\[P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}\]
In the context of the color blindness problem, we use Bayes' theorem to reverse the conditional probability 鈥� we know the probability of color blindness given gender, but we seek the probability of being a certain gender given that one is color-blind. This application illustrates the versatility and analytical power of Bayes' theorem in solving real-world problems.

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Most popular questions from this chapter

Stores \(A, B,\) and \(C\) have \(50,75,\) and 100 employees, respectively, and \(50,60,\) and 70 percent of them respectively are women. Resignations are equally likely among all employees, regardless of sex. One woman employee resigns. What is the probability that she works in store \(C ?\)

Die \(A\) has 4 red and 2 white faces, whereas die \(B\) has 2 red and 4 white faces. A fair coin is flipped once. If it lands on heads, the game continues with die \(A\); if it lands on tails, then die \(B\) is to be used. (a) Show that the probability of red at any throw is \(\frac{1}{2}\) (b) If the first two throws result in red, what is the probability of red at the third throw? (c) If red turns up at the first two throws, what is the probability that it is die \(A\) that is being used?

A certain organism possesses a pair of each of 5 different genes (which we will designate by the first 5 letters of the English alphabet). Each gene appears in 2 forms (which we designate by lowercase and capital letters). The capital letter will be assumed to be the dominant gene, in the sense that if an organism possesses the gene pair \(x X,\) then it will outwardly have the appearance of the \(X\) gene. For instance, if \(X\) stands for brown eyes and \(x\) for blue eyes, then an individual having either gene pair \(X X\) or \(x X\) will have brown eyes, whereas one having gene pair \(x x\) will have blue eyes. The characteristic appearance of an organism is called its phenotype, whereas its genetic constitution is called its genotype. (Thus, 2 organisms with respective genotypes \(a A, b B, c c, d D, e e\) and \(A A, B B, c c,\) \(D D, e e\) would have different genotypes but the same phenotype. In a mating between 2 organisms, each one contributes, at random, one of its gene pairs of each type. The 5 contributions of an organism (one of each of the 5 types) are assumed to be independent and are also independent of the contributions of the organism's mate. In a mating between organisms having genotypes \(a A, b B, c C\) \(d D, e E\) and \(a a, b B, c c, D d, e e\) what is the probability that the progeny will (i) phenotypically and (ii) genotypically resemble (a) the first parent? (b) the second parent? (c) either parent? (d) neither parent?

An urn initially contains 5 white and 7 black balls. Each time a ball is selected, its color is noted and it is replaced in the urn along with 2 other balls of the same color. Compute the probability that (a) the first 2 balls selected are black and the next 2 are white; (b) of the first 4 balls selected, exactly 2 are black.

Independent flips of a coin that lands on heads with probability \(p\) are made. What is the probability that the first four outcomes are (a) \(H, H, H, H ?\) (b) \(T, H, H, H ?\) (c) What is the probability that the pattern \(T, H, H, H\) occurs before the pattern \(H, H, H, H ?\) Hint for part \((c):\) How can the pattern \(H, H, H, H\) occur first?
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