Calculating Die Roll Probabilities 🎲

what is the probability that you obtain a five three times when a fair die is rolled eight times?

There are two elements to solving this problem.

The first one is how many ways can you get three fives when you roll a die eight times?

So that's 1, 2, 3, 4, 5, 6, 7, 8.

And that is going to be eight.

Choose three, which is eight factorial divided by three factorial multiplied by eight minus three factorial, which is five factorial.

Now, if you don't agree with this or you don't understand why, I'm going to attach a link to this video that explains this further.

So that is the first element.

What this does is that it tells me the number of ways that I can get a five when I roll a die eight times, three times.

Because it's not only getting a five in the first three rolls, I can get a five in the first roll.

And then maybe in the last two rolls, I can get a five in the beginning and then one in the middle one in the end.

I can get no fives until the last three.

So these are all events that satisfy the question.

It's three fives in eight rolls.

How?

However, these fives, these three fives, they can appear in different places.

And this accounts for that part of the problem.

Because for every one of those events represented by this, the probability associated is a five, which is going to be a probability of one over six three times.

And so that's to the power of three multiplied by the remaining ones, which is five rolls not being a five, and that is five over six.

And this happens five times.

So this is to the power of five.

And so the correct answer is a.