The number of failures of a testing instrument from contamination particles on the product is a Poisson random variable with a mean of 0.019 failures per hour. (a) What is the probability that the instrument does not fail in an 8-hour shift(b). What is the probability of at least one failure in a 24-hour day?

a) 85.90% probability that the instrument does not fail in an 8-hour shift.

b) 36.62% probability of at least one failure in a 24-hour day

Step-by-step explanation:

The only information that we have is a mean during an interval. So we use the Poisson distribution solve this question.

We have that the probability of exactly x events is given by the following formula:

In which is the mean.

(a) What is the probability that the instrument does not fail in an 8-hour shift

Mean of 0.019 failures per hour. For 8 hours

.

This probability is P(X = 0). So

85.90% probability that the instrument does not fail in an 8-hour shift.

(b) Probability of at least one failure in a 24-hour day?

Mean of 0.019 failures per hour. For 24 hours

.

Either there are no failures, or there is at least one failure. The sum of the probabilities of these events is decimal 1. So

We want to find . So

In which

36.62% probability of at least one failure in a 24-hour day

2