Question

On the leeward side of the island of Oahu, in a small village, about 74% 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) = 0.740 P(2) = 0.192 P(3) = 0.050 (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.) 0.018 (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

Kindly check explanation

Explanation:

Given that:

Probability distribution of a random variable n :

P(n) = (1 - p)^(n-1) * p

p = 0.74

P(n) = (1 - 0.74)^(n-1) * 0.74

P(n) = (0.74)0.26^n-1

Compute the probabilities that n = 1, n = 2, and n = 3.

n = 1

P(1) = (0.74)0.26^1-1

= 0.74 * 0.26^0

= 0.74

n = 2

P(2) = (0.74)0.26^2-1

= 0.74 * 0.26^1

= 0.74 * 0.26

= 0.192

n = 3

P(3) = (0.74)0.26^3-1

= 0.74 * 0.26^2

= 0.74 * 0.0676

= 0.050

Compute the probability that n ≥ 4.

P(n ≥ 4) = 1 - p(n ≤ 3)

p(n ≤ 3) ; n = 1, n = 2, n = 3

p(n = 1) = 0.74

p(n = 2) = 0.192

p(n = 3) = 0.050

P(n ≥ 4) = 1 - (0.74 + 0.192 + 0.050)

P(n ≥ 4) = 1 - 0.982

P(n ≥ 4) = 0.018

(d) What is the expected number of residents in the village you must meet before you encounter the first person of Hawaiian ancestry?

Expected number of residents (μ) :

μ = (1 - p) / p

p = 0.74

μ = (1 - 0.74) / 0.74

μ = 0.26 / 0.74

μ = 0.35