Example 28

EXAMPLE 28 Multiplication rule for $n$ independent events

According to the National Health Interview Survey, 24% of Americans ages 18–44 smoke tobacco. In a random sample of $n = 3$ Americans ages 18–44, find the probability that all three smoke.

Solution

The U.S. Census Bureau estimates that over 100 million Americans are ages 18–44. We will shortly see that, because the sample of 3 is so small compared to the population of over 100 million, it is reasonable to assume that the successive draws are independent. Let $S_{i}$ denote the event that the $i$ th American in the 18–44 age group smokes.

$P (S_{1}) = P (S_{2}) = P (S_{3}) = 0.24$

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Then, using the Multiplication Rule for $n$ Independent Events,

$P (S_{1} and S_{2} and S_{3}) = P (S_{1}) \cdot P (S_{2}) \cdot P (S_{3}) = (0.24) (0.24) (0.24) = {(0.24)}^{3} = 0.013824$

NOW YOU CAN DO

Exercises 93–100.