.
I re-ordered your questions in a way which seems to be more logical to me . . .
On the leeward side of the island of Oahu, in a small village, about 72% of the residents are of Hawaiian ancestry.
Let n = 0, 1, 2, 3, … represent the number of people you must meet until you encounter the first person of Hawaiian ancestry in the village.
(a) 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(0) =
P(1) =
P(2) =
P(3) =
(b) Write out a formula for the probability distribution of the random variable n. (Enter a mathematical expression.)
P(n) =
(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.)
Solution
(a) P(0) = is the probability that the FIRST person you encounter is of Hawaiian ancestry = 0.72.
P(1) = is the probability that the first person you encounter is not of Hawaiian ancestry,
but the SECOND person you encounter is of Hawaiian ancestry = (1-0.72)*0.72 = 0.2016.
P(2) = is the probability that the first TWO persons you encounter are not of Hawaiian ancestry,
but the THIRD person you encounter is of Hawaiian ancestry = (1-0.72)*(1-0.72)*0.72 = 0.056448.
P(3) = is the probability that the first THREE persons you encounter are not of Hawaiian ancestry,
but the FOURTH person you encounter is of Hawaiian ancestry = (1-0.72)*(1-0.72)*(1-0.72)*0.72 = 0.015805.
(b) P(n) = is the probability that the first "n" persons you encounter are not of Hawaiian ancestry,
but the (n+1)-th person you encounter is of Hawaiian ancestry = .
Obviously, the values P(0), P(1), P(2), P(3) form a geometric progression starting from 0.72 with the common ratio of (1-0.72) = 0.28.
(c) P(n>=4) = 1 - P(1) - P(2) - P(3) = 1 - 0.72 - 0.2016 - 0.056448 - 0.015805 = 0.006147.
(d) This Math expectation is
M = 0*P(0) + 1*P(1) + 2*P(2) + 3*P(3) + . . . = + + + . . . =
= =
(see the formula at https://math.stackexchange.com/questions/1269008/formula-for-r2r23r3-nrn)
= = = 0.388888...
Solved.
All questions are answered.
You may round my results (my numbers) by any way you want.