# A pair of fair six-sided dice is thrown eight times. Find the probability that a score greater than #7# is scored no more than five times?

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Before we get into the question itself, let's talk about the method for solving it.

Let's say, for instance, that I want to account for all the possible results from flipping a fair coin three times. I can get HHH, TTT, TTH, and HHT.

The probability of H is

For HHH and for TTT, that is

For TTH and HHT, it's also

When I sum up these results, I get

Notice that if I set

and so in this example, we get:

Now we can do the problem.

We're given the number of rolls as 8, so

Out of 36 possibilities, 15 rolls give a sum greater than 36, giving a probability of

With

We can write out the entire sum of possibilities - from getting all 8 rolls being a sum greater than 7 all the way to getting all 8 rolls being a sum of 7 or less:

but we're interested in summing up only those terms that have our greater than 7 sum happening 5 times or less:

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