Geometric Distributions Consider a game of monopoly.

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Presentation transcript:

Geometric Distributions

Consider a game of monopoly

In monopoly, if you go to jail, you must roll doubles to get out How long can you expect to be in jail?

To get out of jail, you must roll a pair p(pair) = 1/6 q(not a pair) = 5/6 Generate the probability distribution that will display the player getting out on the first roll, or the second roll, or the third role, …

This probability distribution will be controlled by the number of failures. Once a success has been reached, the probability is calculated.

To use a GD model The trials must have 2 outcomes The probabilities do not change The random variable for a GD is the waiting time, (the number of unsuccessful trials before success occurs).

Calculate the PD for getting out of jail in Monopoly in x rolls of the dice X: the number of failed rolls ( in jail ) p (getting doubles [ out of jail ]) = 1/6 q = 5/6 n = ? …. x = 0,1,2,3,4…..

XeventP(x) = x 0Doubles(1/6) 1Fail, Doubles(5/6)(1/6) 2F,F,D(5/6) 2 (1/6) 3F,F,F,D(5/6) 3 (1/6)

Probability in a Geometric Distribution P(x) = q x p Where p is the probability of success in each single trial and q is the probability of failure.

Expected Value E(X) = q / p E(wait time) = = = X Worst case scenario, for 6 rolls, you wait 5, then the 6 th is the successful roll…

Examine examples 3 and 4 on page 393 together

Homework Pg 394 1,2a 3,5,7,9,11,13