Name: Owen

Who is asking: Other Level: All

Question:
This probability question has been bugging me for a while. Two ordinary dice are rolled. If it is known that one shows a 5, what is the probability that they total 8?

I have two different but (seemingly) correct solutions.

The first and most logical one is that if there is 5 showing, then there's only one possible 'show' to make 8 - and that's 3. There are six sides to a dice, and therefore your answer would be one in six.

But consider all the possible combinations that involve one die showing 5.

(1,5);(2,5);(3,5);(4,5);(5,1);(5,2);(5,3);(5,4);(5,5);(5,6);(6,5)for a total of 11 combos. Now there are only two of those that add to 8, being (3,5) and (5,3). Therefore the probability being 2 in 11.

But at the same time I can't discount the other solution. It's driving me nuts!!!!

The second solution is correct.

The flaw in the first solution is that in it we do not distinguish the dice. If you assume that one die is red and the other is green then in the first solution we do not take in account which die shows 5 - red or green.

Now for a paradox, consider the following game: I throw the red and green dices and if they total 8 then I pay you 9 dollars and 10 cents, but with any other total you pay me 2 dollars. Should you accept? Certainely not, since your expected winnings are 9.1*5/36 - 2*31/36 which is negative.

So, you decline, but I am very stubborn, and I will keep the offer open up to the point where you actually see the result of the throw. I throw the dices in a bucket, and I don't let you see them but I tell you that at least one shows a 5. Should you accept the bet now? Well your expected winnings are now 9.1*2/11 - 2*9/11 which is positive. Therefore it looks like you should now accept.

However suppose that before you accept, you ask to name the colour of a dice which shows 5, and I answer "red". According to your first calculation, your odds of winning are one in six, so your expected winnings drop to 9.1*1/6 - 2*5/6 which is negative. Therefore it seems that you should now refuse. But this makes no sense: When you asked the question, you knew that you would get "red" or "green" as an answer. But in both cases your expected value becomes negative, so it seems that just asking the question changes your expected earnings. What is the problem here?