Question

Te probability is 0.5 that an artist makes a craf item with satisfactory quality. Assume the production of each craf item by this artist is independent. What is the probability that at most 3 attempts are required to produce a craf item with satisfactory quality?

Answer 1

The probability that at most 3 attempts are required to produce a craft item with satisfactory quality is 0.9375

Explanation:

Let E be a random variable denoting the event that an artist makes a craft item with satisfactory level.

Then the random variable E follows a Geometric distribution.

A Geometric distribution is defined as the number of failures (k) before the first success.

The probability function of Geometric distribution is:

`P(X=k)=(1-p)^(k)p`, p = Probability of success and k = 0, 1, 2, 3...

The probability of success is, p = 0.5 and the number of failures is, k = 3.

Compute the probability of at most 3 attempts before the first success is:

`P(X\leq 3) =P(X=3)+P(X = 2)+P(X=1) +P(X = 0)\\=[(1-0.5)^(0)*0.5]+[(1-0.5)^(1)*0.5]+[(1-0.5)^(2)*0.5]+[(1-0.5)^(3)*0.5]\\=0.9375`

Therefore, the probability that at most 3 attempts are required to produce a craft item with satisfactory quality is 0.9375.