General multiplication rule example: independent events | Probability & combinatorics

We're told that Maya and Doug are finalists in a crafting competition. For the final round, each of them spins a wheel to determine what star material must be in their craft. Maya and Doug both want to get silk as their star material. Maya will spin first, followed by Doug. What is the probability that neither contestant gets silk?

Pause this video and think through this on your own before we work through this together.

All right, so first let's think about what they're asking. They want to figure out the probability that neither gets silk. So, I'm going to write this in shorthand. I'm going to use "MNS" for Maya no silk. We are also thinking about Doug not being able to pick silk. So, Maya no silk and Doug no silk.

We know that this could be viewed as the probability that Maya doesn't get silk. She, after all, does get to spin this wheel first. Then we can multiply that by the probability that Doug doesn't get silk, Doug no silk, given that Maya did not get silk. Maya no silk.

Now, it's important to think about whether Doug's probability is independent or dependent on whether Maya got silk or not. So, let's remember Maya will spin first, but it's not like if she picks silk that somehow silk is taken out of the running. In fact, no matter what she picks, it's not taken out of the running. Doug will then spin it again, and so these are really two independent events.

So, the probability that Doug doesn't get silk given that Maya doesn't get silk is going to be the same thing as the probability that just Doug doesn't get silk. It doesn't matter what happens to Maya.

So, what are each of these? Well, this is all going to be equal to the probability that Maya does not get silk. There are six pieces or six options of this wheel right over here. Five of them entail her not getting silk on her spin, so five over six.

Then similarly, when Doug goes to spin this wheel, there are six possibilities. Five of them are showing that he does not get silk, Doug no silk. So, times five over six, which is of course going to be equal to twenty-five over thirty-six. And we're done.