Question

The probability that a person over 60 years of age in a certain community drinks alcohol is 2/5 and the probability that a person over 60 years of age has heart disease is 2/15.The probability that a person over 60 years of age drinks alcohol and has heart disease is 1/16.Are "drinking alcohol" and "heart disease" independent events?

Answer 1

Answer on Question #73770 – Math – Statistics and Probability

Question

The probability that a person over 60 years of age in a certain community drinks alcohol is 2/5 and the probability that a person over 60 years of age has heart disease is 2/15.

The probability that a person over 60 years of age drinks alcohol and has heart disease is 1/16. Are "drinking alcohol" and "heart disease" independent events?

Solution

Events A and B will be independent by definition if

$$P(A \text{ and } B) = P(A) * P(B).$$

In our case the probability of event A := "a person over 60 years of age in a certain community drinks alcohol" is 2/5.

The probability of event B := "a person over 60 years of age has heart disease" is 2/15. If they are independent, then

$$P(A \text{ and } B) = P(A) * P(B) = \frac{2}{5} * \frac{2}{15} = \frac{4}{75}$$

But the probability that "a person over 60 years of age drinks alcohol and has heart disease" is 1/16. As long as $4/75 \neq 1/16$, they are not independent.

Answer: "drinking alcohol" and "heart disease" are not independent events.

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