# Probability of rolling a five and a prime pair of six sided dice

This is a silly question, but I can't quite put my finger on where my reasoning is wrong.

Given a pair of 6 sided dice, what is the probability of rolling a 5 and a prime?

My reasoning:

Probability of rolling a five = 1/6

Probability of rolling a prime = 3/6

Probability of first rolling a five and then a prime = 1/6*3/6 = 3/36

Probability of first rolling a prime and then a five = 3/6*1/6 = 3/36

Probability of rolling of rolling a five and a prime = 3/36+3/36 = 6/36

I think where my answer is wrong is that 5 is a prime itself and rolling (5,5) is double counted here.

Is there a better way of doing this type of problem to avoid making these mistakes in the future?

Thank you very much

• The probabilities "rolling a five" and "rolling a prime" are not independent, so you cant express it like a multiplication. May 22 '16 at 17:41
• You are right that (5, 5) is doubled counted. You need to have disjoint sets in order to sum their probabilities. Otherwise, $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. May 22 '16 at 17:41
• You should learn inclusion-exclusion principle, and when you are breaking down the union of events, you will always need to consider the intersection in between to avoid double count issue. Only when you are dealing with disjoint / mutually exclusive events you have a simpler sum.
– BGM
May 22 '16 at 17:42
• Your analysis is correct. We can either remove the double counting by subtracting $\frac{1}{36}$, or treat $5$ as special, so $(2,5)$, $(3,5)$, $(5,2)$, $(5,3)$, $(5,5)$. I don't like the wording, since one could argue that if we got say $(4,5)$ one has rolled a $5$ and a prime. May 22 '16 at 17:43

This is simply a matter of writing explicitly what are the outcomes which satisfy the condition: they are exactly the pairs $$(5,5),(5,3),(5,2),(3,5),(2,5).$$ (You have always to obtain the 5 and the other number must be a prime, i.e. 2,3 or 5). So there are 5 possibilities out of 36. By the way, your final comment is the right explanation.