Question

A manufacturer knows that on average 10​% of toasters produced require repairs within 1 year after they are sold. When 18 toasters are randomly​ selected, find the smallest number x and largest number y such that​ (a) the probability that at least x of them will require repairs is less than 0.54​; ​(b) the probability that at least y of them will not require repairs is greater than 0.8.

Answer 1

a) 3

b) 15

Explanation:

We can treat this as a binomial distribution because there is a fix number of trials (18), the trials are independent, there are only two possible outcomes for each trial and the probability of obtaining a toaster that required repairs is always 0.10.

The formula for the binomial distribution is shown in the picture.

P(0) = 18! ÷ (18 - 0)! ÷ 0! × 0.1⁰ × 0.9¹⁸ = 0.1500

We can do this for each value or we can use a binomial distribution calculator. The number of trials N is 18 and the probability p or π is 0.10.

a)

We find that the probability of obtaining at least 2 toasters that required repairs (x ≥ 2) is 0.5497. Since this value is greater than 0.54 we conclude than the number x is 3.

b)

We can calculate the binomial distribution for the toasters that didn't require repairs instead. In this case the number of trials is going to be the same but the probability in this case is going to be 0.90

The probability that at least 15 of them do not require repairs is 0.9018.