General multiplication rule example: dependent events | Probability & combinatorics

We're told that Maya and Doug are finalists in a crafting competition. For the final round, each of them will randomly select a card without replacement that will reveal what the star material must be in their craft. Here are the available cards. I guess the star material is the primary material they need to use in this competition. Maya and Doug both want to get silk as their star material. Maya will draw first, followed by Doug.

What is the probability that neither contestant draws silk? Pause this video and see if you can work through that before we work through this together.

All right, now let's work through this together. So the probability that neither contestant draws silk—so that would be, I'll just write it another way: the probability that I'll write MNS for Maya, no silk. So Maya, no silk, and Doug, no silk. That's just another way of saying, what is the probability that neither contestant draws silk?

And so this is going to be equivalent to the probability that Maya does not get silk, Maya no silk, right over here. Times the probability that Doug doesn't get silk, given that Maya did not get silk—given Maya, no silk. This line right over this vertical line, this is shorthand for given.

And so let's calculate each of these. So this is going to be equal to the probability that Maya gets no silk. She picked first; there's six options out of here, five of them are not silk. So it is five over six.

And then the probability that Doug does not get silk, given that Maya did not get silk. So if Maya did not get silk, then that means that silk is still in the mix. But there's only five possibilities left because Maya picked one of them, and four of them are not silk. There's still silk as an option.

It's important to recognize that the probability that Doug gets no silk is dependent on whether Maya got silk or not, so it's very important to have this given right over here. If these were independent events—if Maya picked and then put her card back in, and then Doug were to pick separately—then the probability that Doug gets no silk given that Maya got no silk would be the same thing as the probability that Doug gets no silk regardless of what Maya was doing.

And so this will end up becoming four over six, which is the same thing as two thirds.