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

Question

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 answer: 6/36

Correct answer: 5/36

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

Answer 1

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.