Chapter 6.3, Problem 23P

Expand Your Knowledge: Geometric Probability Distribution: Sociology

Geometric Distribution

Suppose we have an experiment in which we repeat binomial trials until we get our first success, and then we stop. Let n be the number of the trial on which we get our first success. In this context, n is not a fixed number. In fact, n could be any of the numbers 1, 2, 3, and so on. What is the probability that our first success comes on the nth trial? The answer is given by the geometric probability distribution.

On the leeward side of the island of Oahu, in the small village of Nanakuli, about 80% of the residents are of Hawaiian ancestry (Source: The HonoluluAdvertiser). 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) Compute the probabilities that $n=1,n=2,andn=3.$(c)Compute the probability that $n\ge 4.$

Hint: $p(n\ge 4)=1-p(n=1)-p(n=2)-p(n=3)$

(d) What is the expected number of residents in Nanakuli you must meet before you encounter the first person of Hawaiian ancestry? Hint: Use $\mu$ for the geometric distribution and round.