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Geometric Distributions

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Explore the application of geometric distributions using Monopoly's jail scenario. In Monopoly, players must roll doubles (1/6) to exit jail, which involves a sequence of trials where success is rolling a double. This study generates a probability distribution to visualize the likelihood of getting out on the first, second, or third roll, based on the number of failed attempts before success. The concepts of success, failure, and expected waiting time provide insights into the behavior of probabilities in games, revealing the nature of geometric distributions in practical scenarios.

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Geometric Distributions

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  1. Geometric Distributions

  2. Consider a game of monopoly

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

  4. 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, …

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

  6. 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).

  7. 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…..

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

  9. Expected Value E(X) = q / p E(wait time) = 1 X = = 5 1 Worst case scenario, for 6 rolls, you wait 5, then the 6th is the successful roll…

  10. Examine examples 3 and 4 on page 393 together

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

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