Dice rolling, and probability.

Question

Oliver rolls three fair standard six-sided dice. What is the probability that there is at least one pair of dice whose top faces sum to $6$? Express your answer as a common fraction.

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I started by saying that there were $6^3$ total possibilities. For the next step, there are $5$ ways to roll a pair of dice and get $6$ when adding them. There is a third dice, so we multiply by $6$ to get $\frac{30}{216}$ but this is wrong! What did I do wrong? Am I on the right track? Or should I do something else to start? (BTW, yes I did simplify the fraction)

Answer 1

I suggest you draw a probability tree. On any toss, the probability of a 6 is 5/36. The experiment ends if you get a 6.

Answer 2

Although you figured it out already, here's why your answer didn't work. (At least in the form you left it)

You artificially created an ordering of the dice. You have shown the possibilities of dice 1 and dice 2 summing to $6$ , but you need to also think about dice 1 and dice 3 summing to $6$, as well as dice 2 and dice 3.

So this will give $30+30+30=90$ ways but then you need to remember inclusion/exclusion to remove all the possibilities you over counted.