## 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 loca

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?

## Answers ( )

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

b) E(x) = 2.5

c) Var(x) = 2.918

Step-by-step 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