Example:
Consider again the game of Craps. The player rolls a pair of
die. If on the first roll of the dice she throws a number other
than 2, 3, 7, 11, or 12, the number she throws is the point.
The rules say she must roll the dice again and continue to
roll
until she throws the point and wins, or a 7, and loses. Say
the point is 4. Based on the probability model for a pair of
dice,
on any given roll following the first:
P(win) = P(x = 4) = 3/36.
P(lose) = P(x = 7) = 6/36.
P(roll again) = 1 – P(win) – P(lose) = 27/36 = 3/4.
For each roll, the game either terminates with probability
1/4, or the player must roll again with probability 3/4.
We model this situation with the Bernoulli
distribution. Take the two outcomes as “roll again”
and “terminate”, and arbitrarily assign the value
0 to the roll again outcome and the value 1 to the terminate
outcome.
For the example
p= P(terminate) = P(1) = 1/4 and
1p = P(roll again) = P(0) = 3/4.
We use these probabilities in the discussion
for the Geometric Distribution.
