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Chapter 6: Problem 69

According to data from the U.S. State Department, the percentage of Americans who have a passport has risen dramatically. In 2007 , only \(27 \%\) of Americans had a passport; in 2017 that percentage had risen to \(42 \%\). Assume that currently \(42 \%\) of Americans have a passport. Suppose 50 Americans are selected at random. a. Find the probability that fewer than 20 have a passport. b. Find the probability that at most 24 have a passport. c. Find the probability that at least 25 have a passport.

Short Answer

Expert verified
The probabilities calculated for parts a, b, and c will be the solution. Note that calculation requires using the binomial distribution formula or statistical software or calculator with binomial distribution functionality.

Step by step solution

01

Understanding the Data

The provided data tells that \(42 \%\) of Americans have a passport. So, the probability (p) of selecting an American with a passport is \(0.42\) . The size of the randomly selected sample (n) is 50.
02

Calculate for part a

We've to find the probability that fewer than 20 Americans have a passport, meaning we calculate for \(k = 0\) to \(k = 19\). This is calculated by summing up the probabilities for each k. Use the binomial probability formula.
03

Calculate for part b

We've to find the probability of at most 24 Americans having a passport. Meaning, we sum the probabilities for \(k = 0\) to \(k = 24\). Use the binomial probability formula.
04

Calculate for part c

Here, the task is to find the probability of at least 25 Americans having a passport. It means we need to sum up the probabilities from \(k = 25\) to \(k = 50\). Alternatively, you can use the complement rule, i.e. one subtracted by the sum of probabilities from \(k = 0\) to \(k = 24\).

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