Suppose a certain ophthalmic trait is associated with eye...

Suppose a certain ophthalmic trait is associated with eye color. Three hundred randomly selected individuals are studied, with results given in the following table. a. What is the probability that a person selected at random has blue eyes? b. What is the probability that a person selected at random has the trait? c. Are events $A$ (has blue eyes) and $B$ (has the trait) independent? Justify your answer. d. How are the two events $A$ (has blue eyes) and $C$ (has brown eyes) related-independent, mutually exclusive, complementary, or all-inclusive? Explain why or why not each term applies.

Suppose a certain ophthalmic trait is associated with eye color. Three hundred randomly selected individuals are studied, with results given in the following table.
a. What is the probability that a person selected at random has blue eyes?
b. What is the probability that a person selected at random has the trait?
c. Are events $A$ (has blue eyes) and $B$ (has the trait) independent? Justify your answer.
d. How are the two events $A$ (has blue eyes) and $C$ (has brown eyes) related-independent, mutually exclusive, complementary, or all-inclusive? Explain why or why not each term applies.

Key Concepts

Probability

Probability is the measure of the likelihood that an event will occur. In this context, it involves calculating the chance that a randomly selected individual exhibits a particular characteristic by comparing the favorable outcomes to the total number of outcomes. This fundamental concept allows us to quantify uncertainty and is central to statistics and risk assessment.

Contingency Tables

A contingency table, or cross-tabulation, is a matrix format used to display the frequency distribution of variables. It is an essential tool in statistics for summarizing data, exploring relationships between categorical variables, and calculating probabilities such as marginal, joint, and conditional probabilities.

Independence of Events

Independence of events refers to a situation in probability where the occurrence of one event does not affect the probability of the occurrence of another event. Testing for independence often involves checking whether the joint probability of two events equals the product of their individual probabilities, which is crucial in determining if events influence one another.

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur at the same time. When two outcomes are mutually exclusive, the occurrence of one event precludes the possibility of the other occurring. This concept is important in probability calculations because the probability of two mutually exclusive events happening together is zero.

Complementary Events

Complementary events are pairs of events where the occurrence of one event implies the non-occurrence of the other, and together they cover the entire sample space. The sum of their probabilities is always equal to one. This concept helps in simplifying probability calculations by considering all possible outcomes.

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