Question

Use the geometric probability distribution to solve the following problem.

On the leeward side of the island of Oahu, in a small village, about 71% of the residents are of Hawaiian ancestry. Let n = 1, 2, 3, … represent the number of people you must meet until you encounter the first person of Hawaiian ancestry in the village.

(a) Write out a formula for the probability distribution of the random variable n. (Enter a mathematical expression.)
P(n) =

(b) Compute the probabilities that n = 1, n = 2, and n = 3. (For each answer, enter a number. Round your answers to three decimal places.)
P(1) =
P(2) =
P(3) =

(c) Compute the probability that n ≥ 4. Hint: P(n ≥ 4) = 1 − P(n = 1) − P(n = 2) − P(n = 3). (Enter a number. Round your answer to three decimal places.)

(d)What is the expected number of residents in the village you must meet before you encounter the first person of Hawaiian ancestry? Hint: Use μ for the geometric distribution and round. (Enter a number. Round your answer to the nearest whole number.)
residents

Answer 1

Let N be the random variable that denotes the number of people you meet until you encounter the first person of Hawaiian ancestry in the village.

About 71% of the residents are of Hawaiian ancestry. Therefore, p = 0.71

(a) Answer :

Random variable N follows the geometric distribution with parameter p = 0.71

N Geometric (p = 0.71)

The probability distribution of N is

P(n) = p * (1 - p)n-1 ; n = 1, 2, .........

        = 0              ; otherwise

Therefore, the probability distribution of N is

P(n) = 0.71 * 0.29n-1 ; n = 1, 2, 3, ........

        = 0                      ; otherwise

(b) Answer :

P(1) = 0.71 * 0.290 = 0.710

Therefore, P(1) = 0.710

P(2) = 0.71 * 0.291 = 0.206

Therefore, P(2) = 0.206

P(3) = 0.71 * 0.292 = 0.060

Therefore, P(3) = 0.060

(c) Answer :

P(n 4) = 1 - P(n < 4)

              = 1 - (P(n = 1) + P(n = 2) + P(n = 3))

              = 1 - P(n = 1) - P(n = 2) - P(n = 3)

              = 1 - 0.710 - 0.206 - 0.060

              = 0.024

Therefore, P(n 4) = 0.024

(d) Answer :

E(N) = = 1 / p

               = 1 / 0.71

               = 1.409

               1

Therefore, the expected number of residents in the village you must meet before you encounter the first person of Hawaiian ancestry is 1.