# Non-transitive dice probabilities

I got the following question on non-transitive dice, as shown in the following picture.

Suppose you roll two white dice against two red dice. What is the probability that the sum of the white dice is greater than the sum of the red dice?

And the answer is as follows.

Can someone explain to me why the total number of possible outcomes is here 144? I know that if we would have 4 normal dice, the total number of outcomes would be 1296, hence my question. I also don't understand the probabilities where the white dice beat the red dice since I have calculated a probability of 9/36, 18/36, and 9/36 for 4, 7, and 10 respectively, when rolling the 2 white dice.

Thanks!

• This is not your doing but rather that of the person who wrote the material you quote, yet I cannot let this go without some note of protest. While I realize that "dice" as singular (die) has been used so regularly it's now listed as an acceptable term for a lone die, it seems even more bizarre to also use the distinctly singular die for the plural. Next of course, people start to add an 's' back to die because it doesn't sound plural enough (I've seen it), and on it goes - we have errors leading to still more errors and soon nobody has any idea how many dice we meant to refer to. Commented Jan 17, 2023 at 21:44

## 1 Answer

This is merely a choice on how far to simplify fractions. In this case the author has inconsistently chosen to use simplest individual fractions for the white dice, but no simplification anywhere else.

If the author had not simplified fractions at all, then the final denominator would indeed be 1296. Situations like this are why I generally do not recommend simplifying fractions when doing combinatorics.

• That makes sense, thanks! So for example the probability for getting a 6 (upper left corner) would be 225/1296. How does one calculate that? Commented Jan 17, 2023 at 20:32
• Each cell probability is the product of the corresponding row and column probability, following the rule for independent events. So in this case we have 25/36 (red) times 9/36 (white). Commented Jan 17, 2023 at 21:12