Counting principle and factorial
Let's say that I've just won some type of contest at a car dealership, and they're going to give me a brand new car. And in deciding which car they give me, they're first going to randomly select the engine type. So the engine will come in two different varieties. It'll either be a four-cylinder or a six-cylinder engine. And they're literally just going to flip a fair coin to decide whether I get a four-cylinder engine or a six-cylinder engine. Then they're going to pick the color. And there's four different colors that the cars come in. So I'll write color in a neutral color. So you could get a red car. That's not red. Let me do that in actual red color, or close to red. You could get a red car, you could get a blue car, you could get a green car, or you could get a white car. And once again, they're going to have the red, blue, green, and white in little slips of paper in a bowl and they're just going to pick one of them out. So all of these are equally likely. So given this, that they're just going to flip a coin to pick the engine, and that all of these, the colors all equally likely, I want to think about the probability of getting a six-cylinder white car. So I encourage you to pause the video and think about it on your own. Well, one way to think about this is, well, what are all the equally likely possible outcomes? And then which of those match six-cylinder white car? Well, first, we could think about the engine decision. We're either going to get a four-cylinder engine. So the first decision is the engine. You could view it that way. You're either going to get a four-cylinder engine, or you're going to get a six-cylinder engine. Now, if you got a four-cylinder engine, you're either going to get red, blue, green, or white. And if you've got a six-cylinder engine, once again, you're either going to get red, blue-- I think you see where this is going. That's not blue. Red, blue, green, or white. So how many possible outcomes are there? Well, you could just count. You could kind of say, the leaves of this tree diagram-- one, two, three, four, five, six, seven, eight possible outcomes. And that makes sense. You have two possible engines times four possible colors. You see that right here-- one group of four, two groups of four. So this outcome right here is a four-cylinder blue car. And this outcome over here is a six-cylinder green car. So there's eight equally possible outcomes. And which outcome matches the one that we, I guess, are hoping for, the white six-cylinder car? Well, that's this one right over here. It's one of eight equally likely events. So we have a 1/8 probability. This wasn't the only way that we could have drawn the tree diagram. We could have thought about color as the first row of this tree. So we could have said, look, we're either going to get a-- let me do it down here, so I have a little more space-- we're either going to get a red, a blue-- that's not blue. Changing colors is the hard part. A blue, a green, or a white car. And then for each of those colors, I'm either going to get a four-cylinder or a six-cylinder engine. So it's either going to be four or six. This would be another way of drawing a tree diagram to represent all of the outcomes. So what is this outcome right over here? This is a six-cylinder red car. This is a four-cylinder blue car right over here. Which is the one that we care about? White six-cylinder car? That's this outcome right over here. Once again, you see you have eight equally likely outcomes. And that happens because you have four possible colors. And for each of those four possible colors, you have two different engine types.