You roll two fair $6$-sided dice. Given at least one of the rolls is a $4$, what is the probability that the sum is even?

I feel like one could approach this problem using probability laws or Bayes' Rule. However, I thought of something that is much simpler, but could be wrong.

If at least one of the rolls is a $4$, in order for the sum to be even, the other die would have to show a $2,4,$ or $6.$ In other words, an even number would have to be rolled.

The probability of rolling an even number is $\frac{1}{2}$, which is my final answer.

Thanks for reading this short post. Any thoughts?


There are five outcomes in the joint event "a four shows and the sum is even": $$\{(4,2),(4,4),(4,6),(2,4),(6,4)\}$$

How many outcomes form the conditioning event: "a four shows"?


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