Question

Two dice are rolled together and X is the highest score minus the lowest score of the dice. What are the possible values of X?

Answer 1

If you consider just positive number, you have 15 possible values, if you also consider negatives, including zero, you have 31 possible values, with zero occurring with higher probability.

Explanation:

First of all, this question should be reformulated, if you have two dice, what is the meaning of higher and lower scores if they are just two? it seems meaningless. I am going to answer the question:

Two dice are rolled together and X is the difference of the scores of the dice. What are the possible values of X?

Consider two independent dice; it means that the probability of one does not affect the other, mathematically:

`Pr(D1|D2)=Pr(D2|D1)=Pr(D1)*Pr(D2)` , this property is mandatory for the upcoming calculations:

Case 1: positive and negative numbers are valid
In this case, we have:

`Sum= Dice1 + Dice2`, since they are independent events, we have

`6*5=30` outputs plus 1 of zero that occurs when the numbers repeat themselves; which occurs with probability`6/36=1/6=16%` .

Case 2: * just positive numbers are valid*

In this case, we have, since the order does matter:

`Sum= max(DiceRoll) + min(DiceRoll)`

`(6,2)=(6!)/(2!4!)=15`, plus 1 for zero.

Conclusions

To know the values, just take the maximum, `6 - 1= 5`, and minimum `1- 6 = - 5`. Thus `omega =[-5,5]`. For the second case: `omega =[0,5]`. To associate probabilities, just see how many times they occurs, e.g. Five can occur for: 6-1, just, for the second case, for both, 1-6, therefore 2 times.

See

Dice
Combinations