Thursday, April 14, 2016

Analysis of Two Dice Throws

Analysis of Two Dice Throws

The table below lists each outcome from two dice throws.  Not all outcomes are unique.   For example, there are two ways to get a one on one dice and a two on the other dice.   The one could be on the first dice and the two on the second dice or the order could be reversed.

Questions:

·      List all unique outcomes from the two dice roll experiments. 

  • ·      What is the probability of each outcome?


  • ·      What is the probability of getting snake eyes?   (Snake eyes is 1 on each dice.)  How does the probability of getting snake eyes compare to the probability of getting (1,2).


  • ·      There are six outcomes where the first dice is equal to the second dice.   What is the probability the throw of the two dice results in two identical numbers?


  • ·      There are 15 outcomes where the two dice throws result in two different values.   What is the probability that the values of the two dice throws differ in value?



Outcomes from two throw of the Dice
First Dice
Second Dice
Probability of Each Outcome
1
1
1
0.0278
2
2
1
0.0278
3
3
1
0.0278
4
4
1
0.0278
5
5
1
0.0278
6
6
1
0.0278
7
1
2
0.0278
8
2
2
0.0278
9
3
2
0.0278
10
4
2
0.0278
11
5
2
0.0278
12
6
2
0.0278
13
1
3
0.0278
14
2
3
0.0278
15
3
3
0.0278
16
4
3
0.0278
17
5
3
0.0278
18
6
3
0.0278
19
1
4
0.0278
20
2
4
0.0278
21
3
4
0.0278
22
4
4
0.0278
23
5
4
0.0278
24
6
4
0.0278
25
1
5
0.0278
26
2
5
0.0278
27
3
5
0.0278
28
4
5
0.0278
29
5
5
0.0278
30
6
5
0.0278
31
1
6
0.0278
32
2
6
0.0278
33
3
6
0.0278
34
4
6
0.0278
35
5
6
0.0278
36
6
6
0.0278


Answer:   In most games like Risk or Monopoly it does not matter if one throws two dice and gets (i,j) or (j,i).   For instance in monopoly if you throw (1,2) or (2,1) you move three.   Similarly, both of these throws in Risk give the player a max value of 2 and a min value of 1.

Below I have listed all unique outcomes from two throws of the dice and the probability of each outcome.


Unique Outcomes From Two Dice Throws Regardless of Dice Outcomes
Outcome #
Unique Pair
Probability
1
1
1
0.0278
2
1
2
0.0556
3
1
3
0.0556
4
1
4
0.0556
5
1
5
0.0556
6
1
6
0.0556
7
2
2
0.0278
8
2
3
0.0556
9
2
4
0.0556
10
2
5
0.0556
11
2
6
0.0556
12
3
3
0.0278
13
3
4
0.0556
14
3
5
0.0556
15
3
6
0.0556
16
4
4
0.0278
17
4
5
0.0556
18
4
6
0.0556
19
5
5
0.0278
20
5
6
0.0556
21
6
6
0.0278
1.0000



Note the probability of getting snake eyes (1,1) is 1/32 while the probability of getting unique pair (1,2) is 1/18.   This is because there is only one way to get snake eyes but two ways to get (1,2).   (The only way to get snake eyes is to get 1 on both throws.   The two ways to get (1,2) are one on the first dice and two on the second dice and 2 on the first dice and 1 on the second dice.)

There are six unique outcomes when the dice are equal  -- (1,1), (2,2), (3,3), (4,4), (5,5) and (6,6).   Each of these outcomes has a probability of 1/36.   The probability that the two die throws results in identical outcomes is 6/36 or 1/6.


There are 15 unique outcomes when the two dice have different values.   (See the table above for each of these outcomes.)   The likelihood of each outcome is 1/18.   The likelihood that one of the 15 outcomes where the dice values differs occurs is 15/18 or 5/6.

The probability that dice values are equal is 1/6.  The complement of equal dice values is the event dice values differ.   The probability of the complement is 5/6.

In the next post I will use what we have learned here about the probability of two dice throws to figure out the likelihood of different outcomes in the game of Risk when the attacking army attacks with two armies and the defending army defends with two armies.


Note in a previous post I looked at what happens in the game of Risk when two attacking armies are greeted by one defending army.

Previous Problem in the Game of Risk:


Authors Note:  I wrote a short book Statistical Applications of Baseball in 1997.  It was well reviewed by the journal Chance.   A number of people have told me they have benefited from this book.   The book is very inexpensive.

Statistical Applications of Baseball




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