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Color blindness is a condition where a person is unable to distinguish between certain colors. The probability of a male being color-blind is 0.042, or 4.2%. This means, if we randomly select a male, there is a 4.2% chance that he will be color-blind.
This probability can be used to predict outcomes in larger groups. If you have a group of 53 men, you can use binomial probability to calculate various probabilities related to the number of color-blind men.
In our exercise, we will determine:

• The probability that exactly 5 men are color-blind.
• The probability that no more than 5 men are color-blind.
• The probability that none are color-blind.
• The probability that at least 1 man is color-blind.

The binomial coefficient, often represented as \(\binom{n}{k}\), is a key part of the binomial probability formula. It calculates the number of ways to choose \(k\) successes from \(n\) trials.
For example, to find the probability that exactly 5 out of 53 men are color-blind, we need the binomial coefficient \( \binom{53}{5} \). You can calculate it as:
\[ \binom{53}{5} = \frac{53!}{5!(53-5)!} \] This computation shows how many ways we can choose 5 color-blind men from 53. Once you have this value, you multiply it by the probability of success raised to the power of \(k\) (number of successes) and the probability of failure raised to the power of \(n-k\) (number of failures).
Learning how to use the binomial coefficient effectively allows you to tackle many different types of probability problems involving fixed numbers of trials and binary outcomes (success/failure).

Cumulative probability sums up the probabilities of multiple outcomes. For instance, if we want to find the probability that no more than 5 men are color-blind in our group of 53, we calculate the cumulative probability for 0, 1, 2, 3, 4, and 5 color-blind men.
Mathematically, this is represented as:
\[ P(X \leq 5) = \sum_{k=0}^{5} \binom{53}{k} (0.042)^k (1-0.042)^{53-k} \] Here, we need to add up the probabilities for each case from 0 to 5. This result helps us to understand the broader range of possible outcomes rather than focusing on a single specific outcome.
Cumulative probabilities are particularly useful when calculating probabilities for ranges of outcomes, such as finding the likelihood that 'at most' or 'at least' a certain number of events will occur. By adding the probabilities for all these individual cases, we obtain a more comprehensive understanding of the scenario.