Question

1. Suppose I work on a factory line doing quality control work. Since I’ve been

working here for such a long time, I am fairly confident that 15% of all items

(independent of all other items) need to be sent back to be re-worked. These items

take a fairly long time to check properly, but it is the same set of steps every time,

so I always check 25 items every shift. Let X be the number of items sent back to be re-worked during a given shift.

What is the distribution of X? Give the name of the distribution and the appropriate parameters.

What is the mean and variance of this distribution? Give your answer as a number, but include the formulas (or logic) used.

What is the probability I send back less than 4 items to be re-worked during a

single shift?

What is the probability I send back exactly 3 items to be re-worked during a single

shift?

Answer 1

a) It is a binomial distribution.

n = 25

p = 0.15

b) mean = np = 25 * 0.15 = 3.75

    variance = np(1 - p)

                  = 25 * 0.15 * (1 - 0.15)

                  = 3.1875

c) P(X = x) = nCx * p^x * (1 - p)^{n - x}

P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

             = 25C0 * (0.15)^0 * (0.85)^25 + 25C1 * (0.15)^1 * (0.85)^24 + 25C2 * (0.15)^2 * (0.85)^23 + 25C3 * (0.15)^3 * (0.85)^22 = 0.4711

d) P(X = 3) = 25C3 * (0.15)^3 * (0.85)^22 = 0.2174