Question

6. In the Pirates of the Aegean video game, you may have to investigate 0, 1, 2, 3, 4, or 5 grottoes without success before you finally locate a treasure chest. You conclude that the probability you will have to investigate at mostxgrottoes without success is , and F(5) = 1. (Remember that is the number of permutations(ordered lists withoutreplacement) of kitems from npossibilities.)a) What is the probability you will have to investigate exactly 3 grottoes without success?b) What is the expected numberE(X) of grottoes you will investigate without success?c) What is the variance Var(X) of the number of grottoes you will investigate without success?

Answer 1

a) probability of investigating exactly 3 grottoes = 1/6

b) E(x) = 2.5

c) Var(x) = 2.918

Explanation:

Since there are 6 cases in the distribution, the number of elements in the sample space is 6

a) probability of investigating exactly 3 grottoes = 1/6

b) Expected value E(x) = ∈(xP(x))

E(x) = (0*1/6) + (1*1/6) + (2*1/6) + (3*1/6) + (4*1/6) + (5*1/6)

E(x) = 2.5

c) variance of x, Var(x) = ∈ [(E(x) - x)² * P(x)]

Var(x) = [(2.5-0)²*1/6] + [(2.5-1)²*1/6] + [(2.5-2)²*1/6] + [(2.5-3)²*1/6] + [(2.5-4)²*1/6] + [(2.5-5)²*1/6]

Var(x) = 1.042 + 0.375 + 0.042 + 0.042 + 0.375 + 1.042

Var(x) = 2.918