Question

A manufacturing company performs a quality-control analysis on the ceramic tile it produces. Suppose a batch of 22 tiles has 7 defective tiles. If 5 tiles are sampled at random, what is the probability that exactly 1 of the sampled tiles is defective? How many ways can 5 tiles be selected from 22 tiles? ways (Type a whole number.)

Answer 1

The probability that exactly one of the five sampled tiles is defective is calculated using the hypergeometric distribution. The number of ways to select five tiles from a batch of 22 is calculated using combinations.

Step-by-step explanation:

To find the probability that exactly 1 of the 5 sampled tiles is defective, we need to use the hypergeometric distribution formula, which is suitable for scenarios without replacement. The formula is:

P(X = k) = [ (C(g, k) * C(N-g, n-k)) / C(N, n) ], where:

Here, N = 22, g = 22 - 7 = 15, n = 5, and k = 1. Thus, the probability P(X = 1) is:

P(X = 1) = [ C(7, 1) * C(15, 5-1) ] / C(22, 5)

Calculate the combinations:

• C(7, 1) = 7
• C(15, 4) = 1365
• C(22, 5) = 26,334

So, P(X = 1) = (7 * 1365) / 26,334, which simplifies to the probability value.

To find out how many ways 5 tiles can be selected from 22 tiles:

Total ways = C(22, 5) = 26,33