Use the geometric probability distribution to solve the
following problem.
On the leeward side of the island of Oahu, in a small village,
about 81% 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
(a)
p = 0.81
The probability distribution of the random variable n,
P(n) =
= , n = 1,2,3,.....
(b)
P(1) = 0.810
P(2) = 0.154
P(3) = 0.029
(c) P(n ≥ 4)
= 1 - P(n = 1) - P(n = 2) - P(n = 3)
= 0.007
(d) The expected number of residents in the village you must meet before you encounter the first person of Hawaiian ancestry = 1/p = 1/0.81 = 1.235
= 1.235
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