The definition of probability is the number of occurrences that meet a specified criteria (the size of the event space) divided by the total number of possibilities (the size of the sample space.)

There are 36 different rolls of a pair of dice. Note that a roll of (1,3) is different from a roll of (3,1). Imagine that the two dice are rolled one at a time, or that the two dice are different colors. Thus order matters except in the case of doubles (e.g. (1,1) where red 1 green 1 is the same as green 1 red 1.)

There are 3 ways to roll dice with a sum of 4: (1,3),(3,1), and (2,2)

There are 5 ways to roll dice with a sum of 6: (1,5),(5,1),(2,4),(4,2),(3,3)

Thus the size of the sample space is 36 while the size of the event space is 3+5=8.

**Thus the probability of rolling a 4 or 6 is 8/36 or 2/9.**

(Probability is usually written as a fraction in lowest terms, so 2/9. Occasionally it is written as a decimal so in this case ` 0.bar(2) ` , or as a percent; here approximately 22.2%.)

Assuming that the two dice are thrown together (and thus the "order" that the dice are thrown/land don't matter), there are 21 total pairs of dice values that can occur. They are as follows:

`{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,2),(2,3),(2,4),(2,5),(2,6),(3,3),(3,4),(3,5),(3,6),(4,4),(4,5),(4,6),(5,5),(5,6),(6,6)}`

We must now find how many of these pairs sum to 4 or 6. These are:

`{(1,3),(2,2),(1,5),(2,4),(3,3)}`

So the probability of rolling a sum of 4 or 6 is `5/21`

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