By Wen Jun
Learning Objectives: Discovering the memoryless property for Geometric Distributions
 
Interactive Prompt:  
 
[Prompt generated using ChatGPT]
Simulate two players, Alex and Ben, rolling a fair die until they get their first 6.

 
Alex is starting fresh.

Ben has already rolled 10 times without success.

 
Let X X be the number of rolls until the first 6 appears, X tilde operator text Geometric end text not stretchy left parenthesis p equals 1 over 6 not stretchy right parenthesis X ∼  Geometric(p =    6   1​  ).
Run many trials to find how many rolls each player needs.
Plot histograms for Alex’s total rolls (X subscript A X A​ ) and Ben’s additional rolls after 10 failures (X subscript B minus 10 X B​  −  10).

 
Goal:
Show that both distributions are the same — the probability of success on the next roll is always 1 over 6   6   1​  , no matter how many times you’ve already failed.

 
Key Interaction:

 
Adjust number of past failures (slider).

Run 1 or 1000 simulations.

Compare histograms of X subscript A X A​  and X subscript B minus s X B​  −  s.

Display empirical probability of success each roll.