Question

A company that produces fine crystal knows from experience that 17% of its goblets have cosmetic flaws and must be classified as "seconds." (Round your answers to four decimal places.)(a)Among seven randomly selected goblets, how likely is it that only one is a second

Answer 1

0.3891 = 38.91% probability that only one is a second

Explanation:

For each globet, there are only two possible outcoes. Either they have cosmetic flaws, or they do not. The probability of a goblet having a cosmetic flaw is independent of other globets. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

17% of its goblets have cosmetic flaws and must be classified as "seconds."

This means that
`p = 0.17`

Among seven randomly selected goblets, how likely is it that only one is a second

This is P(X = 1) when n = 7. So

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 1) = C_(7,1).(0.17)^(1).(0.83)^(6) = 0.3891`

0.3891 = 38.91% probability that only one is a second