Question: What is the probability that a person starts
the game by throwing doubles on the first throw and then throws a second time
and lands on Pennsylvania Railroad?
(Pennsylvania Railroad is the railroad that is 15 squares away from Go,
the opening square.)
Concepts: In order to do this problem you need to
understand the concepts of probability, mutually exclusive events,
independence, and intersection.
Analysis: Note that the person can’t get to Pennsylvania Railroad with just one throw because the largest sum of the two dice is 12. Under
the rule of Monopoly the person has to throw a doubles on the first throw to
get a second throw. However, the person
can’t get to Penn Ave in two throws by throwing a (1,1) on the first throw
again because the sum of the dice can’t equal 13.
The first throw has to be (2,2), (3,3), (4,4) (5,5) or
(6,6).
There is only one way to get each double; hence, the
probability of each double is (1/36).
The second throws
must respectively sum to 11, 9, 7, 5, 3,
The probability for the sum of two dice was presented in a
previous post.
Briefly:
There are two ways to get to 11 or 3; hence the probability
the sum of two dice =11 is 1/18 and the probability the sum of two dice is
equal to 3 is 1/18.
List for 3 is (1,2) and (2,1).
There are four ways to get 5 or 9 so the probability the sum
of two dice equals 9 is 1/9.
List for 5 is (1,4) (4,1) (2,3) and (3,2).
There are 6 ways to get to 7 ((6,1), (1,6), (5,2), (2,5),
(3,4) and (4,3). The probability of
getting a 7 is 1/6.
There are five paths to 15 but to get to 15 you have to both
land on one of the paths by getting one of five doubles and get the appropriate
sum.
The outcome from the first throw of the two dice is
independent from the outcome of the second throw of the two dice. Hence, the probability that both events
occur is the product of the probabilities for the two outcomes.
The five paths are mutually exclusive so you can add the
five probabilities to get the answer.
Probability of getting to
Penn Railroad on two throws on first turn


First throw

Prob first throw outcome

Sum of dice on second
throw giving a total of 15

Probability for second
throw outcome

Product of two
probabilities

(2,2)

0.027777778

11

0.055555556

0.00154321

(3,3)

0.027777778

9

0.111111111

0.00308642

(4,4)

0.027777778

7

0.166666667

0.00462963

(5,5)

0.027777778

5

0.111111111

0.00308642

(6,6)

0.027777778

3

0.055555556

0.00154321

0.013888889

This calculation suggests that around every 14 out of 1000
turns a person can throw doubles and move a total of 15 squares on the first
and second throw.
One could also get to Penn Railroad on three throws when the
first two are doubles. Interested readers
should make this calculation.
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