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Transcript Document

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…..
X
event
P(x) = x
0
Doubles
(1/6)
1
Fail, Doubles
(5/6)(1/6)
2
F,F,D
(5/6)2(1/6)
3
F,F,F,D
(5/6)3(1/6)
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.
Expected Value
E(X) = q / p
E(wait time) =
5
6
1
6
1
=
5
6
X
1
6
= 5
1
Worst case scenario, for 6 rolls, you wait 5,
then the 6th is the successful roll…
Examine examples 3 and 4 on
page 393 together
Homework
• Pg 394
• 1,2a
• 3,5,7,9,11,13