Problem 49 - A Graphical Approach to Precalculus with Limits

Question

Color-Blind Males The probability that a male will be color blind is \(0.042 .\) Approximate the probabilities that in a group of 53 men, the following are true. A. Exactly 5 are color blind. B. No more than 5 are color blind. C. At least 1 is color blind.

Step-by-Step Solution

Answer

A. Approximately 0.112 B. Approximately 0.922 C. Approximately 0.999

Step 1: Define the Random Variable and Distribution

Let's define the random variable \( X \) as the number of color-blind men in a group of 53 men. Since we're dealing with a fixed number of trials with two possible outcomes (color-blind or not), and each trial is independent, \( X \) follows a Binomial distribution. Thus, \( X \sim \text{Binomial}(n=53, p=0.042) \), where \( n \) is the number of trials, and \( p \) is the probability of success (being color-blind).

Step 2: Calculate Probability of Exactly 5 Color-Blind Men

Using the binomial probability formula \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), we calculate the probability of exactly 5 men being color-blind. Plugging in values: \[ P(X = 5) = \binom{53}{5} (0.042)^5 (0.958)^{48} \]. Compute this to get the numerical probability.

Step 3: Calculate Probability of No More Than 5 Color-Blind Men

To find the probability of no more than 5 men being color-blind, we need to sum the probabilities from 0 to 5 color-blind men: \[ P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) \]. Compute each using the binomial formula and add them together.

Step 4: Calculate Probability of At Least 1 Color-Blind Man

Recall that the probability of at least one success is complementary to the probability of no successes: \( P(X \geq 1) = 1 - P(X = 0) \). Calculate \( P(X = 0) \) using the binomial formula and then find \( P(X \geq 1) \) by subtracting from 1.

Key Concepts

Probability

Probability is a measure of how likely an event is to occur. It's essential in understanding how random variables behave in various scenarios, such as determining the likelihood of a particular outcome in a series of trials.
In this exercise, we examined the likelihood or probability that a certain number of men out of a group would be color-blind. The probability of an individual man being color-blind was given as 0.042. This simple probability needs to be incorporated into a broader scenario where multiple such trials occur.
Using probability we can make significant predictions. For example:

• The probability of exactly 5 out of 53 men being color-blind.
• The combined probability of no more than 5 men being color-blind.
• The probability of at least one man being color-blind.

Each of these probabilities gives us insights into different aspects of our scenario, helping us to understand what outcomes are most likely or unlikely to happen. Probability plays a vital role in managing uncertainty and making informed decisions based on statistical data.

Random Variables

Random variables are crucial in understanding statistical scenarios. They assign numerical values to each outcome of a random process. In simpler terms, they help us quantify uncertainty.
In this exercise, the random variable was defined as the number of color-blind men in a group of 53 men. This number could vary depending on each individual's likelihood of being color-blind. In probability theory, a random variable is typically represented by a symbol such as \( X \).
There are two main types of random variables:

• Discrete random variables - These have countable outcomes. In our example, the number of color-blind men, \( X \), is a discrete random variable since it can only take specific integer values like 0, 1, 2, and so on up to 53.
• Continuous random variables - These represent outcomes that are not countable, often taking any value within a range.

Understanding how random variables work is essential for effectively applying probability to practical problems, allowing us to create models that can predict outcomes based on given probabilities.

Binomial Theorem

The Binomial Theorem and the binomial distribution are fundamental concepts when dealing with experiments that involve multiple trials with two possible outcomes. This is particularly useful for problems like our exercise, where a trial can either result in a man being color-blind or not.
The binomial distribution follows when we satisfy these conditions:

• The number of trials \( n \) is fixed.
• Each trial has only two outcomes: success (being color-blind) or failure (not being color-blind).
• The probability of success \( p \) is constant for each trial.
• The trials are independent of each other.

Given these conditions, the binomial distribution can be used to calculate the probabilities for different outcomes.
The formula for the probability of exactly \( k \) successes in \( n \) trials is:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]Where \( \binom{n}{k} \) is the binomial coefficient representing the number of ways to choose \( k \) successes from \( n \) trials. This dynamic approach allows us to compute not just isolated probabilities, but sums of probabilities, to answer a variety of statistical questions, such as those in our exercise.