Question

Of the bolts manufactured for a certain application, 83% meet the length specification and can be used immediately, 13% are too long and can be used after being cut, and the remaining are too short and must be scrapped. Find the probability that fewer than 9 out of a sample of 10 bolts can be used (either immediately or after being cut). Round the answer to four decimal places.

Answer 1

solution:

Given that

The percentage of bolts that can be used immediately = 83%

The percentage of bolts that can be used after cut = 13%

The probability of bolts that can be used = p = 83% + 13% = 0.96 [ remember both mutually exclusive]

The Total Sample of bolts = n = 10

Let X be the random variable of bolts that can be used

Here, X ~B(10, 0.96)

B(n,X) = nCx * (p) x * (1-p)n-x

P (fewer than 9 bolts can be used ) = P(X<9)

= 1 - P(X>=9) [since, P(X<x) + P(X>=x) = 1 ]

= 1 - [ P(X=9) + P(X=10) ]

= 1 - [  10C9 * (0.96) 9 * (0.04) +  10C10 * (0.96) 10 * (0.04)0 ]

= 1 - [0.277 + 0.665 ]

= 0.0580

Probability that fewer than 9 bolts can be used = 0.0580