Question

Alice is playing a game in which she needs to advance exactly 15 spaces in order to win. The number of spaces that she may advance is determined by rolling a die. What is the probability that she will win in 3 turns?

Answer 1

If you roll a die three times, how many total outcomes are there?

Answer 2

Um...18?

Answer 3

This is the counting principle.
Event 1 has 6 outcomes.
Event 2 has 6 outcomes.
Event 3 has 6 outcomes.
Total number of outcomes = 6 * 6 * 6 = 216.

Answer 4

Ah...ok.

Answer 5

But still, I'm a bit confused.

Answer 6

Now we know that by rolling a die three times, there are a total of 216 outcomes.
Now we need to find how many outcomes have a sum of 15 for the three rolls of the die.

Answer 7

That's what I'm confused about; how do you find that out?

Answer 8

Let's figure it out.
If one roll gives you a 1, can you get a sum of 15?

Answer 9

Nope.

Answer 10

What if one roll is 2?

Answer 11

Still no.

Answer 12

How many ways can you roll a sum of 15?
5 + 5 + 5 = 15

6 + 6 + 3 = 15
6 + 3 + 6 = 15
3 + 6 + 6 = 15

4 + 5 + 6 = 15
4 + 6 + 5 = 15
5 + 4 + 6 = 15
5 + 6 + 4 = 15
6 + 4 + 5 = 15
6 + 5 + 4 = 15

Answer 13

There are 10 ways of getting a sum of 15.

Answer 14

The probability is:

P(sum of 15) = 10/216 = 5/108

Answer 15

Thank you so much for your help on this problem. I know it took up a lot of your time, I'm really sorry. Nonetheless, thank you.

Answer 16

Don't be sorry. You're welcome. I'm glad I was able to help.