In Exercises 1-3, find the indicated probabilities using the geometric distribut
ID: 3305007 • Letter: I
Question
In Exercises 1-3, find the indicated probabilities using the geometric distribution the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities 1. One out of every 100 tax returns that a tax auditor examines requires an audit. Find the probability that (a) the first return requiring an audit is the 25th return the tax auditor examines, (b) the first return requiring an audit is the first or second return the tax auditor examines, and (c) none of the first five returns the tax auditor examines require an audit. (Source: CBS News) 2. Twenty percent of U.S. adults have some type of mental illness. You randomly select six U.S. adults. Find the probability that the number of U.S. adults who have some type of mental illness is (a) exactly two, (b) at least one, and (c) less than three. (Source: U.S. Department of Health and Human Services) 3. The mean increase in the United States population is about four people per minute. Find the probability that the increase in the U.S. population in any given minute is (a) exactly six people, (b) more than eight people, and (c) at 2 most four people. (Source: U.S. Census Bureau)Explanation / Answer
1. Geometric distribution
p = 1/100 = 0.01
a) Required probability = (1-p)^24 * p = (1-0.01)^24 * 0.01 = 0.0079
b) Required probability = p + (1-p)*p = 0.01 + 0.99*0.01 = 0.0199
c) Required probaiblity = (1-p)^5 = 0.99^5 = 0.951
2. Binomial Distribution
p = 0.2 n = 6
a) P(X = 2) = 6C2*p^2*(1-p)^4 = 0.2458
b) P(X>=1) = 1 - P(X=0) = 1 - (1-p)^6 = 0.9415
c) P(X<3) = 0.9011
3. Poisson's Distribution
lambda = 4 per minute
a) P(X = 6) = (=POISSON.DIST(6,4,FALSE)) = 0.1042
b) P(X>8) = 1 - P(X<=8)
P(X <= 8) = 0.9786 (Excel formula =POISSON.DIST(8,4,TRUE))
P(X>8) = 1 - 0.9786 = 0.0214
c) P(X<=4) = 0.6288 ( =POISSON.DIST(4,4,TRUE))