EXAMPLE 16Applying the binomial probability distribution formula

A report from SleepFoundation.org reported that 20% of Americans are sleep-deprived (defined as getting less than six hours sleep per night, on average). This has serious consequences for our nation's highways and productivity. Suppose we take a random sample of four Americans. Find the probability that the following numbers of people are sleep-deprived:

  1. None
  2. At least one
  3. Between one and three, inclusive
  4. Five

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Solution

We apply the steps for solving binomial probability problems.

  • Step 1 We have a random sample of four Americans, so the number of trials is . “Success” is denoted as a particular American being sleep-deprived. The report states that 20% of Americans are sleep-deprived, so and .
  • Step 2 For (a), . For (b), ; that is, . For (c), ; that is, ; that is, . For (d), .
  • Step 3 We apply Step 3 for each of (a)–(d) as follows:

  1. Step 3 To find the probability that none () of the Americans are sleep-deprived, we use the binomial probability formula:

    Therefore, the probability that none of the Americans in the sample are sleep-deprived is 0.4096.

  2. Step 3 Note that “at least one” includes all possible values of except . In other words, the two events () and () are complements of each other. Therefore, from the formula for the probability for complements in Section 5.2 (page 260), we have

    The probability that at least one of the Americans is sleep-deprived is 0.5904.

  3. Step 3 We need to find the probability that either or or of the Americans are sleep-deprived. Because these three values of are mutually exclusive, we find the required probability by using the Addition Rule for Mutually Exclusive Events.

    So we calculate the following:

    Thus, . The probability is 0.5888 that between one and three, inclusive, of the Americans in the sample of four are sleep-deprived.

  4. Step 3 In a binomial experiment, the number of successes can never exceed the number of trials . In other words, , always. So, if our sample has only Americans, then . It is not possible for there to be five Americans who are sleep-deprived.

NOW YOU CAN DO

Exercises 15–28.