Question

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. Twenty-two percent of former smokers say they tried to quit four or more times before they were habit-free. You randomly select 10 former smokers. Find the probability that the first person who tried to quit four or more times is

(a) the third person selected
(b) the fourth or fifth person selected
(c) one of the first seven people selected

Answer 1

(a) The probability that the third person selected is the first to have tried to quit four or more times is calculated using the geometric distribution.

(b) The probability that the fourth or fifth person selected is the first to have tried to quit four or more times is also determined using the geometric distribution.

(c) The probability that one of the first seven people selected is the first to have tried to quit four or more times is found through the binomial distribution.

Step-by-step explanation:

(a) For the geometric distribution, the probability of success (a former smoker who tried to quit four or more times) on any given trial is 22%. The probability that the third person selected is the first success is calculated using the formula
`\( (1-p)^(k-1) * p \)`, where p is the probability of success and k is the trial number.

(b) Similarly, the probability for the fourth or fifth person selected being the first success is calculated using the same geometric distribution formula.

(c) For the binomial distribution, the probability of getting exactly k successes in n trials is given by the binomial probability formula. In this case, it calculates the probability of having the first success within the first seven people selected.