206 CHAPTER 4 Discrete Probability Distributions Finding Binomial Probabilities Using Formulas A survey found that 22% of U.S. adults say that the economy, unemployment, and jobs are the most important problems facing the U.S. today. You randomly select four adults and ask them whether the economy, unemployment, and jobs are the most important problems facing the U.S. today. Find the probability that (1) exactly two of them respond yes, (2) at least two of them respond yes, and (3) fewer than two of them respond yes. (Source: Ipsos) SOLUTION 1. Using n = 4, p = 0.22, q = 0.78, and x = 2, the probability that exactly two adults will respond yes is P122 = 4C210.222 210.7822 = 610.222210.7822 = 0.177. 2. To find the probability that at least two adults will respond yes, find the sum of P122, P132, and P142. Begin by using the binomial probability formula to write an expression for each probability. P122 = 4C210.222 210.7822 = 610.222210.7822 P132 = 4C310.222 310.7821 = 410.222310.7821 P142 = 4C410.222 410.7820 = 110.222410.7820 So, the probability that at least two will respond yes is P1x Ú 22 = P122 + P132 + P142 = 610.2222(0.7822 + 410.222310.7821 + 10.222410.7820 ≈ 0.212. 3. To find the probability that fewer than two adults will respond yes, find the sum of P102 and P112. Begin by using the binomial probability formula to write an expression for each probability. P102 = 4C010.222 010.7824 = 110.222010.7824 P112 = 4C110.222 110.7823 = 410.222110.7823 So, the probability that fewer than two will respond yes is P1x 6 22 = P102 + P112 = 10.222010.7824 + 410.222110.7823 ≈ 0.788. TRY IT YOURSELF 5 The survey in Example 5 found that 16% of U.S. adults say that public health, disease, and illness are the most important problems facing the U.S. today. You randomly select five adults and ask them whether public health, disease, and illness are the most important problems facing the U.S. today. Find the probability that (1) exactly two of them respond yes, (2) at least two of them respond yes, and (3) fewer than two of them respond yes. (Source: Ipsos) Answer: Page A38 You can use technology to check your answers. For instance, the TI-84 Plus screen at the left shows how to check parts 1 and 3 of Example 5. Note that the second entry uses the binomial CDF feature. A cumulative distribution function (CDF) computes the probability of “x or fewer” successes by adding the areas for the given x-value and all those to its left. EXAMPLE 5 TI-84 PLUS binompdf(4, 0.22, 2) .17667936 binomcdf(4, 0.22, 1) .78775632 Study Tip The complement of “x is at least 2” is “x is less than 2.” So, another way to find the probability in part 3 of Example 5 is P1x 6 22 = 1 - P1x Ú 22 ≈1 - 0.212 = 0.788. For help with translating verbal phrases, see Integrated Review at MyLab Statistics
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