# When 2 dice are thrown the sum of the numbers that turns up is 10. What is the probability that one of the dice has a 4. Please explain.

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If the sum of the numbers shown on two dice is 10, what is the probability that one number is a 4?

(1) There are three ways for the sum of the dice to be 10: rolls of 4,6 and 5,5 and 6,4.

The sample space has three items, and the event space has two items as there are two possible throws with a 4, Thus the probability is 2/3

(2) We can use conditional probability to confirm:

`P(B|A)=(P(A"and"B))/(P(A)) `

Here we define P(A) as the probability of rolling a sum of 10, and P(B) as the probability that a die shows a 4.

P(A and B) is the probability that the sum is 10 and there is a 4. This is 2/36 as there are 36 possible rolls with (4,6) and (6,4) the only rolls whose sum is 10 with a 4.

P(A) is the probability of rolling a 10. This is 3/36 as there are 36 possible rolls, three of which sum to 10. (4,6),(5,5),(6,4).

Thus the probability of a pair of dice having a 4 if we know the sum is 10 is

`P(B|A)=(2/36)/(3/36)=2/3 `

**Sources:**

A die is a cube and there are 6 numbers, {1,2,3,4,5,6}, that can turn up when the die is thrown.

When two two dice are thrown the sum of the numbers that turn up is 10. This is the case when the following sets of numbers turn up on the dice {(4,6)(5,5),(6,4)}. There are total of three cases in which the sum is 10 and in two of these the number on one of the die is 4. The required probability is therefore 2/3.

If the sum of the numbers that turn up when 2 dice are thrown is 10, there is a probability of 2/3 that the number on one of the dice is 4.

A lot of it has to do with what perspective you are looking at.

If you consider the chart on the webpage given, there are 36 combination of rolls you can get from two dice. When you consider the sum being 10, there are only 3 combinations. So, the probability of getting a 10 would be 3/36 = 1/12. But, then, of those 3 combinations, only 2 of them have a 4. So, then, the probability of getting a sum of 10 with one die being a 4 would be 2/36 = 1/18, when taking into consideration all possible combinations.

But, then, if you consider the perspective of taking into consideration all possible combinations of 10 only, then the probability would be 2/3, 3 total combinations of 10, with 2 of them having a 4.

**Sources:**

Dice have 6 faces with the numbers 1-6 on them. If you have two dice that add up to a sum of 10, the possible combinations would be {4, 6}, {6, 4}, and {5, 5}. Out the combinations, there are 2 that include a 4, out of 3 total possibilities. Your probability would be `(2)/(3)`.