Sanderson M. Smith

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 mx = 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 TI-83.
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.

 x Probability (to 4 decimals) (xi)(pi) (to 4 decimals) 1 0.3800 0.0038 2 0.2356 0.4712 3 0.1461 0.4382 4 0.0906 0.3623 5 0.0562 0.2808 6 0.0348 0.2089 7 0.0216 0.1511 8 0.0134 0.1071 9 0.0083 0.0747 10 0.0051 0.0514 11 0.0032 0.0351 12 0.0020 0.0237 13 0.0012 0.0159 14 0.0008 0.0106 15 0.0005 0.0071 16 0.0003 0.0047 17 0.0002 0.0031 18 0.0001 0.0020 19 0.0001 0.0013 20 0.0000 0.0009 21 0.0000 0.0006 22 0.0000 0.0004 23 0.0000 0.0002 24 0.0000 0.0002 25 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 mx = 1/p avoids all of the computation represented in the third column above.