When we roll one die, we have a 1 in 6 probability of getting any particular num

Question

When we roll one die, we have a 1 in 6 probability of getting any particular number on the die.   When we roll a pair of dice, there are 36 different pairs that can be produced, yet only 11 actual distinct values.

Explain how the probability associated with the roll of each individual die in the pair explains the higher variability in the total outcome of the roll of each pair. How do the concepts of permutations and combinations apply to this example? Discuss how the notion of degree of freedom can be used to illustrate the accumulating results of a set of dice rolls.

Explanation / Answer

When rolling two dice, distinguish between them in some way: a first one and second one, a left and a right, a red and a green, etc. Let (a,b) denote a possible outcome of rolling the two die, with a the number on the top of the first die and b the number on the top of the second die. Note that each of a and b can be any of the integers from 1 through 6. Here is a listing of all the joint possibilities for (a,b):

Assuming we have a standard six-sided die, the odds of rolling a particular value are 1/6. There is an equal probability of rolling each of the numbers 1 to 6. But, when we have two dice, the odds are not as simple.

Sample Space - (1,1)(1,2)(1,3)(1,4)(1,5)(1,6)(2,1)(2,2)(2,3)(2,4)(2,5)(2,6)(3,1)(3,2)(3,3)(3,4)(3,5)(3,6)(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

This total number of possibilities can be obtained from the multiplication principle: there are 6 possibilities for a, and for each outcome for a, there are 6 possibilities for b. So, the total number of joint outcomes (a,b) is 6 times 6 which is 36. The set of all possible outcomes for (a,b) is called the sample space of this probability experiment.

Let's count how many ways there are to get each value, 2 through 12

OUTCOME LSIT OF COMBINATIONS TOTAL PROBABILITY

2 - 1 + 1 - 1 1/36 = 2.78%
3 - 1 + 2, 2 + 1 - 2 2/36 = 5.56%
4 - 1 + 3, 2 + 2, 3 + 1 - 3 3/36 = 8.33%
5 - 1 + 4, 2 + 3, 3 + 2, 4 + 1 - 4 4/36 = 11.11%
6 - 1 + 5, 2 + 4, 3 + 3, 4 + 2, 5 + 1 - 5 5/36 = 13.89%
7 - 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2, 6 + 1 -6 6/36 = 16.67%
8 - 2 + 6, 3 + 5, 4 + 4, 5 + 3, 6 + 2 - 5 5/36 = 13.89%
9 - 3 + 6, 4 + 5, 5 + 4, 6 + 3 - 4 4/36 = 11.11%
10 - 4 + 6, 5 + 5, 6 + 4 - 3 3/36 = 8.33%
11 - 5 + 6, 6 + 5 - 2 2/36 = 5.56%
12 - 6 + 6 - 1 1/36 = 2.78%

THE SUM OF ALL TEH PROBABILITIES IS 1

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