Question

Use the geometric probability distribution to solve the following problem.

On the leeward side of the island of Oahu, in a small village, about 72% 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

Answer 1

p = 0.72 72% are Hawaiin ancestry     
n number of people to meet until first person of Hawaiian ancestry is met     
  that is (n-1) people we meet are non-Hawaiian and 1 person is Hawaiian ancestry     
       
a) Let X be the number of persons to meet until the first Hawaiian ancestry person is met      
X follows a Geometric distribution and the probabilities are given by      

b) To find P(1), P(2), P(3)      
We use Excel function NEGBINOM.DIST with number of success value = 1   to get Geometric Distribution probability  
P(1) = NEGBINOM.DIST(0, 1, 0.72, FALSE)      
          = 0.720      
P(1) = 0.720     
       
P(2) = NEGBINOM.DIST(1, 1, 0.72, FALSE)      
          = 0.202      
P(2) = 0.202      
       
P(3) = NEGBINOM.DIST(2, 1, 0.72, FALSE)      
          = 0.056      
P(3) = 0.056      
       
       
c) P(n ≥ 4) = 1 - P(n < 4)      
             = 1 - [P(n = 1) + P(n = 2) + P(n = 3)]      
             = 1 - (0.72 + 0.202 + 0.056)      
             = 0.022      
P(n ≥ 4) = 0.022      
       
d) The Expected value of a Geometric Distribution is       
= 1/p      
= 1/0.72      
= 1.39      
= 1 (Rounding to nearest whole number)     
The Expected value of a Geometric Distribution is 1