Dice and probability, v2

Last month I wrote about probability.

Here's the problematic way of thinking: if I have a 1/3 chance of rolling a 1 or a 5 with one die, then I have a 2/3 chance of rolling a 1 or a 5 with two dice.

I looked at a 6x6 image of possible dice combinations (on Ed Collins' Web site) that shows all 36 two-dice combination possibilities. If you count, you'll find that 24 of the 36 dice represented are indeed 1s or 5s. 24 out of 36...that's 2/3.

So why isn't the probability 2/3?

Because probability is about dice rolls, two dice taken together, and sometimes the 1s and 5s appear together. Four times, to be exact. Which means that the number of successful rolls is 20/36 (or 55.5 out of 100). Better than half, but not nearly 2/3 that my faulty math (1/3 + 1/3 = 2/3) might imply.

What I find fascinating is that my initial thought (1/3 + 1/3 = 2/3) has some merit--look at the chart, the numbers 1 & 5 do appear 2/3 of the time. What wasn't accurate was what I was measuring, which is combined dice rolls that score, and not the total number of individually thrown dice.