Answer to Question #142789 in Statistics and Probability for Yala96

Question

A safety engineer feels that 35% of all industrial accidents in her plant are caused by failure of employees to follow instructions. She decides to look at the accident reports (selected randomly with replacement) until she finds one that shows an accident caused by failure of employees to follow instructions.

(i) On average, how many reports would the safety engineer expect to look at until she finds a report showing an accident caused by employee failure to follow instructions? (Enter your answer rounded to 4 decimal places)

(ii) What is the probability that the safety engineer will find the first report showing an accident caused by employee failure to follow instructions within the first 3 examined reports? (Enter your answer rounded to 4 decimal places)

Answer 1

Let X = the number of accidents the safety engineer must examine until she finds a report showing an accident caused by employee failure to follow instructions. Then X is a geometric random variable. All three characteristics are met. Each accident report she reads is a Bernoulli trial: the accident was either caused by failure of employees to follow instructions or not. She would need to read at least one accident report before she stops.

(i) The mean of a random variable following the geometric distribution is

$$E[X]=\mu=\dfrac{1}{p}$$

Given $p=0.35$

Then

$$E[X]=\mu=\dfrac{1}{0.35}\approx2.8571$$

(ii)

$$P=p+(1-p)p+(1-p)^2p=$$

$$=0.35+(1-0.35)(0.35)+(1-0.35)^2(0.35)=$$

$$=0.725375\approx0.7254$$