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 write out a formula for the probability distribution of the random variable n, we need to consider the probability of encountering the first person of Hawaiian ancestry at each step.

In the village of Nanakuli, 80% of the residents are of Hawaiian ancestry. Therefore, the probability of encountering the first person of Hawaiian ancestry on the nth step is given by the probability of not encountering any person of Hawaiian ancestry in the first n-1 steps, multiplied by the probability of encountering the first person of Hawaiian ancestry on the nth step.

The probability of not encountering any person of Hawaiian ancestry in the first n-1 steps is (1 - 0.8)^(n-1) = 0.2^(n-1). The probability of encountering the first person of Hawaiian ancestry on the nth step is 0.8.

Thus, the probability distribution of the random variable n can be expressed as:

P(n) = 0.2^(n-1) * 0.8

(b) To compute the probability that n = 1, we substitute n = 1 into the formula from part (a):

P(1) = 0.2^(1-1) * 0.8 = 0.8

To compute the probability that n = 2, we substitute n = 2 into the formula from part (a):

P(2) = 0.2^(2-1) * 0.8 = 0.16

To compute the probability that n = 3, we substitute n = 3 into the formula from part (a):

P(3) = 0.2^(3-1) * 0.8 = 0.032

(c) To compute the probability that n is greater than or equal to 4, we need to sum the probabilities for all n values greater than or equal to 4.

P(n >= 4) = P(4) + P(5) + P(6) + ...

Since the probability of each n is calculated using the formula from part (a), we can write:

P(n >= 4) = 0.2^(4-1) * 0.8 + 0.2^(5-1) * 0.8 + 0.2^(6-1) * 0.8 + ...

This is an infinite geometric series with a common ratio of 0.2 * 0.8 = 0.16. The sum of an infinite geometric series with a common ratio less than 1 can be calculated using the formula:

Sum = a / (1 - r)

where 'a' is the first term and 'r' is the common ratio. In this case, a = 0.2^3 * 0.8 and r = 0.16.

P(n >= 4) = (0.2^3 * 0.8) / (1 - 0.16)

(d) In Waikiki, it is estimated that about 4% of the residents are of Hawaiian ancestry. Following the same steps as parts (a), (b), and (c), we can calculate the probability distribution, probabilities for n = 1, n = 2, n = 3, and the probability that n is greater than or equal to 4 using the appropriate values for the probability of encountering the first person of Hawaiian ancestry (0.04) and the probability of not encountering any person of Hawaiian ancestry (1 - 0.04).

(a) The probability distribution of the random variable n can be written as P(n=k), where k represents the number of people you must meet until you encounter the first person of Hawaiian ancestry in the village of Nanakuli. Since 80% of the residents are of Hawaiian ancestry, the probability of meeting a person of Hawaiian ancestry on any encounter is 0.8 (or 80%) and the probability of not meeting a person of Hawaiian ancestry is 0.2 (or 20%). Therefore, the probability distribution can be represented as follows:

P(n=k) = (0.2)^(k-1) * 0.8

(b) To find the probability that n equals a specific value, we substitute the value of k into the formula derived in part (a). Let's calculate the probabilities:

P(n=1) = (0.2)^(1-1) * 0.8 = 0.8

P(n=2) = (0.2)^(2-1) * 0.8 = 0.16

P(n=3) = (0.2)^(3-1) * 0.8 = 0.032

(c) To compute the probability that n is greater than or equal to 4, we need to calculate the probability of not encountering a person of Hawaiian ancestry in the first three encounters and subtract it from 1:

P(n>=4) = 1 - [P(n=1) + P(n=2) + P(n=3)]
= 1 - (0.8 + 0.16 + 0.032)
≈ 1 - 0.992
≈ 0.008

So, the probability that n is greater than or equal to 4 is approximately 0.008 or 0.8%.

(d) In Waikiki, it is estimated that about 4% of the residents are of Hawaiian ancestry. Using the same approach as in parts (a), (b), and (c), the probability distribution, probabilities, and probability that n is greater than or equal to 4 in Waikiki can be calculated as:

(a) Since 4% (or 0.04) of the residents are of Hawaiian ancestry:

P(n=k) = (0.96)^(k-1) * 0.04

(b) Calculating the probabilities:

P(n=1) = (0.96)^(1-1) * 0.04 = 0.04

P(n=2) = (0.96)^(2-1) * 0.04 = 0.0384

P(n=3) = (0.96)^(3-1) * 0.04 = 0.036864

(c) Calculating the probability that n is greater than or equal to 4:

P(n>=4) = 1 - [P(n=1) + P(n=2) + P(n=3)]
= 1 - (0.04 + 0.0384 + 0.036864)
≈ 1 - 0.115264
≈ 0.884736

So, the probability that n is greater than or equal to 4 in Waikiki is approximately 0.884736 or 88.47%.

a) p(1-p)^n-1