Question

The
nodules for customers to crack open. Suppose they have 350 nodules and 250 geodes to randomly draw 12 rocks from with which to make these bags. Let X be the number of geodes in a bag.

mineral museum in Butte, MT is interested in selling bags of rocks containing a mix of geodes and

(a) [2 pts] What distribution does X follow? Include the parameters for the identified distribution in proper notation.

(b) [3 pts] How many geodes can be expected in any given bag? What is the variance of the number of geodes in a bag?

(c) [2 pts] Calculate the probability that exactly half of the rocks in a randomly selected bag are geodes.

(d) [2 pts] Calculate the probability that more than one third of the rocks in a randomly selected bag are

geodes.

Answer 1

a) X follow hypergeometric dstribution.

N = 350 + 250 = 600

k = 250

n = 12

b) Expected value = n * k / N = 12 * 250 / 600 = 5

Variance = n * k * ( N - k ) * ( N - n ) / [ N² * ( N - 1 ) ] = 12 * 250 * (600 - 250) * (600 - 12) / [600² * (600 - 1)] = 2.86

c) P(X = 6) = 250C6 * 350C6 / 600C12 = 0.1919

d) P(X > 4) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4))

                  = 1 - (350C12 / 600C12 + 250C1 * 350C11 / 600C12 + 250C2 * 350C10 / 600C12 + 250C3 * 350C9 / 600C12 + 250C4 * 350C8 / 600C12)

                  = 1 - 0.3903

                  = 0.6097