Wednesday, May 11, 2016

Monopoly PR5: Probability of moving 15 squares on one turn

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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