Sanderson M. Smith
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GEOMETRIC SETTING EXAMPLE
Suppose that your probability of winning a game is 38%, and that each game is independent of any other game. Let x = the number of games played before you win. Then x is a random variable with possible values 1,2,3,4,... . This is a geometric setting with p = .38, and the mean m_{x} = 1/p = 1/.38 = 2.6316.
The probability that x = 1 is 0.38.
The probability that x = 2 is (.62)(.38) = 0.2356.
The probability that x = 3 is (.62)^{2}(.38) = 0.146072.
etc., etc.
The Excel chart above displays the probability distribution of the random variable x.
The table below shows probability values for x = 1,2,3,...,24,25. The probabilities of getting values greater than 25 are very small. Decimals are shown to four decimal places.
Side note: It is very easy to get these values on your TI83.
Highlight list L1 and use seq(x,x,1,25,1).
Highlight list L2 and use geometpdf(.38,L1).
To get the product column, highlight list L3 and type L1*L2.The probability and product columns represent rounded figures calculated on a spreadsheet that did the computations using many more decimal places.




0.3800 
0.0038 

0.2356 
0.4712 

0.1461 
0.4382 

0.0906 
0.3623 

0.0562 
0.2808 

0.0348 
0.2089 

0.0216 
0.1511 

0.0134 
0.1071 

0.0083 
0.0747 

0.0051 
0.0514 

0.0032 
0.0351 

0.0020 
0.0237 

0.0012 
0.0159 

0.0008 
0.0106 

0.0005 
0.0071 

0.0003 
0.0047 

0.0002 
0.0031 

0.0001 
0.0020 

0.0001 
0.0013 

0.0000 
0.0009 

0.0000 
0.0006 

0.0000 
0.0004 

0.0000 
0.0002 

0.0000 
0.0002 

0.0000 
0.0001 
TOTALS 
1.0000 
2.6314 

This column total is, in reality, just a teensy bit short of 1. Remember that there are an infinite number of x values. 
This column total is the mean of the random variable x. Note that this very close to the theoretical mean obtained by using the formula 1/p = 1/.38. 
If you have a geometric setting, a simple formula m_{x} = 1/p avoids all of the computation represented in the third column above.
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