SOLUTION: On the leeward side of the island of Oahu, in a small village, about 76% of the residents are of Hawaiian ancestry. Let n = 1, 2, 3, … represent the number of people you must mee

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Question 1157266: On the leeward side of the island of Oahu, in a small village, about 76% 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) = ?


Answer by Edwin McCravy(20066)   (Show Source): You can put this solution on YOUR website!
Theoretically you may never encounter a person of Hawaiian ancestry, but that is
very unlikely, but we treat this as an infinite tree diagram.
"Yes" means the person we meet is of Hawaiian ancestry, and "no" means they
aren't.


      0.24   0.24   0.24   0.24   0.24   0.24   0.24 ... = (0.24)n+
start—————no—————no—————no—————no—————no—————no————— ...
     \      \      \      \      \      \      \     ...
      \0.76  \0.76  \0.76  \0.76  \0.76  \0.76  \    ...
       \      \      \      \      \      \      \   ... 
        yes    yes    yes    yes    yes    yes       ... 


The probability that we will meet such a person the 1st time is
0.76

The probability that we will not meet such a person until the 2nd time 
we meet somebody is (0.24)(0.76)

The probability that we will not meet such a person until the 3rd time
we meet somebody is (0.24)2(0.76)

The probability that we will not meet such a person until the 4th time
we meet somebody is (0.24)3(0.76)
...
...
...
The probability that we will not meet such a person until the nth time
we meet somebody is (0.24)n-1(0.76)

 n   P(n)
---  ----
 1   0.76
 2   0.24(0.76)  
 3   0.242(0.76)
 4   0.243(0.76)
 .
 .
 .
 n   0.24n-1(0.76)

P(n) = 0.24n-1(0.76)  

Edwin
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