Number Properties Probability Pg. 73

[BryanG505]

For question 1, it reads, "What is the probability that the sum of two number cubes will yield a 10 or lower?".

I understand that the best approach would be to do 1- P(Not A), so 1 - P(11,12). I know that it is 6 X 6 for the denominator.

For the numerator, would there not be 4 possible combinations: 6-5, 6-6, 5-6, 6-6? Are you not supposed to double count that 6-6 because it technically is the other die that rolls a 6 first? If you were to write out all possible combinations for this (1-6, 1-5, 1-4 ...), if you do not count the ones that are double rolled the same number (1-1, 2-2, 3-3...), then you are creating a grid of 5 X 6 with only 30 possible outcomes instead of the 36.

If you do double count the 6s, then should it not be 4/36 = 1/9, so 1-(1/9) = 8/9 chance of it being 10 or lower.

Thank you

[Sage Pearce-Higgins]

This is from the problem set at the end of Chapter 5 of the Number Properties Strategy Guide.

You don't need to double count the (6, 6) outcome: this is only one of the 36 possible outcomes, unlike the (5, 6) outcome which can be achieved with one die being 5, and the other 6, and vice versa.

To make this clear, write out a 6 by 6 grid showing all the 36 possible rolls. By the way, what's the most likely sum of the the two dice? (This is a useful answer to know for anyone playing Monopoly.)