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?

Apr 19, 2016

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:

$P r \left(D 1 | D 2\right) = P r \left(D 2 | D 1\right) = P r \left(D 1\right) \cdot P r \left(D 2\right)$ , this property is mandatory for the upcoming calculations:

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

$S u m = D i c e 1 + D i c e 2$, since they are independent events, we have

$6 \cdot 5 = 30$ outputs plus 1 of zero that occurs when the numbers repeat themselves; which occurs with probability6/36=1/6=16% .

Case 2: * just positive numbers are valid*

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

$S u m = \max \left(D i c e R o l l\right) + \min \left(D i c e R o l l\right)$

(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 $\omega = \left[- 5 , 5\right]$. For the second case: $\omega = \left[0 , 5\right]$. 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.

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