On the leeward side of the island of Oahu, in the small village of Nanakuli, about 80% 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 of Nanakuli.
(a) Write out a formula for the probability distribution of the random variable n.
(b) Computer the probability that n=1, n=2, n=3.
(c) Compute the probability that n is greater than or equal to 4.
(d) In Waikiki it is estimated that about 4% of the residents are of Hawaiian ancestry. Repeat parts (a) (b) and (c) for Waikiki.
(a) To find the probability distribution for the random variable n, we need to assess the probability of meeting a person of Hawaiian ancestry at each possible encounter.
Let's denote P(n) as the probability that the first person of Hawaiian ancestry will be met at the nth encounter. Since there is an 80% chance of each person encountered in Nanakuli being of Hawaiian ancestry, and assuming independence between encounters, we can express the probability distribution as:
P(n) = (0.8)^(n-1) * 0.2
(b) To compute the probability for specific values of n:
For n = 1:
P(1) = (0.8)^(1-1) * 0.2 = 0.2
For n = 2:
P(2) = (0.8)^(2-1) * 0.2 = 0.16
For n = 3:
P(3) = (0.8)^(3-1) * 0.2 = 0.128
(c) To compute the probability that n is greater than or equal to 4:
P(n ≥ 4) = 1 - P(n < 4)
= 1 - (P(1) + P(2) + P(3))
= 1 - (0.2 + 0.16 + 0.128)
= 1 - 0.488
= 0.512
So, the probability that n is greater than or equal to 4 in Nanakuli is 0.512.
(d) Assuming that 4% of the residents in Waikiki are of Hawaiian ancestry, the probability distribution and calculations are as follows:
P(n) = (0.04)^(n-1) * 0.96
For n = 1:
P(1) = (0.04)^(1-1) * 0.96 = 0.96
For n = 2:
P(2) = (0.04)^(2-1) * 0.96 = 0.0384
For n = 3:
P(3) = (0.04)^(3-1) * 0.96 = 0.001536
To compute the probability that n is greater than or equal to 4:
P(n ≥ 4) = 1 - P(n < 4)
= 1 - (P(1) + P(2) + P(3))
= 1 - (0.96 + 0.0384 + 0.001536)
= 1 - 0.999936
= 0.000064
So, the probability that n is greater than or equal to 4 in Waikiki is 0.000064.