A pair of dice are rolled. It is known that the sum of those equals to 6

Question

A pair of dice are rolled. It is known that the sum of those equals to 6. What is the probability that the product of two dices is equal to 5 or 9?

Answer 1

For a problem this small-scale, counting the possibilities is the easiest option. To roll a sum of 6, there are exactly five possible rolls: (1,5), (2,4), (3,3), (4,2), and (5,1) (denoting a roll as (first dice, second dice)). Three of those meet the product requirement, so the final answer is $\frac{3}{5}$.

Note that you do not count (3,3) twice, because there are not two distinct situations that produce that result. In combinatorics, when double-counting is necessary, it can only be necessary because there are two possible circumstances yielding the same result. That usually means that you're not counting the right things anyway.

Answer 2

It will help to write the sample space of the experiment of rolling a die and having a sum equalling $6$ as: {$(1,5) (2,4) (3,3) (4,2) (5,1)$}. Out of this two cases {$(1,5) (5,1)$} satisfy the product of $5$ and one case {$(3,3)$} with a product of $9$. The total number of cases with a product of either $5$ or $9$ is $3$.

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Hence, we have a probability (using conditional probability) of $$\frac{3}{5}$$

Answer 3

For sum as 6, Sample space = {$(1,5) (2,4) (3,3) (4,2) (5,1)$} $= 5$

For product as 5 or 9, Favourable case = {$(1,5) (3,3) (5,1)$} $= 3$

Probability = $\frac{3}{5}$