Multiplication rule for independent events
- [Voiceover] Let's say that you're on some type of a game show and you've been doing quite well. And you're now at the round where you get to pick your fantabulous vacation. And so there are three possible places that you could go. You could go on an island beach vacation. Island beach vacation. You could go skiing on a ski vacation. Or, you could go camping. Now those aren't the only possibilities because for each of those vacations there's different amount of time that you could go on them. So you could go for one day. You could go for two days. Two days. Or you could go for three days. Three. Put that in a different color. You could go for three days. You could go for three days. So the first question I'd wanna know is, well what is the-- They're gonna randomly pick either a one day ski vacation or a two day island vacation. The first question I wanna know is, what are all of the possible outcomes here? What is the sample space? What is the space from which we are going to pick your particular vacation package? Well for the sample space, we can construct a grid. Which you can see that I've essentially been constructing while I wrote down all of the possibilities. So let me draw out the sample space with these uneven looking grid lines. All right, I think you get the picture. And I'll just abbreviate it. You can go on a one day, a one day island trip. This one I, this is one day island trip. You can go on a two day, two day-- Actually, let me just write it this way. All of these are going to be one day. Right? Because of the one day column. All of these are going to be two days. Two days. Two days. And all of these are going to be three days because it's on the three day column. And all of the ones in this row are going to be island trips. So it's one day island trip, two day island trip, three day island trip. This second row, it's all ski trips. One day ski trip, two day ski trip, three day ski trip. And then finally everything in this third row, they're campng trips. One day camping trip, two day camping trip, three day camping trip. So just like that, we have constructed the sample space right over here. You see that there's one, two, three, four, five, six, seven, eight, nine outcomes. And let's say that each of these outcomes are on a little piece of paper and they put it in a barrel and they roll it up. And for our purposes we can assume that they are all equally likely outcomes. So we're gonna assume equally likely outcomes. So if we do assume equally likely outcomes, we can figure out a probability. Maybe you live in someplace that's cold and you're really not in the mood to go skiing. In fact, you'd like to spend several days away from the snow. So let's ask ourselves a question. What is the probability that you're going to win something at least two days on a vacation without snow. Two days on vacation without snow. You're gonna randomly pick one of these nine outcomes. What's the probability that it's going to be at least-- It's gonna give you a vacation that gives you at least two days without snow. Well, let's just think a little bit about it. We know the sample space and we know each of the outcomes are equally likely. There are nine equal outcomes here. So let's write that down. We got nine equal outcomes. Now how many of the outcomes satisfy this? This event, this constraint. At least 2 days of vacation-- Let me write this. Two days. Without snow. Whether it falls or touching it or whatever. So you're essentially avoiding skiing. You want at least two days on something other than skiing. And we're assuming you're not gonna go camping in some type of alpine. You're camping in some place that's warm. Well, let's think about these outcomes. So this one is no snow but it's only one day. This is two days without snow so we can circle that one. This is three days without snow so we can circle that one. All of these have snow. This is one day without snow so we're not gonna do this one. This is two days without snow and this is three days without snow. And so four of the equally likely outcomes satisfy this constraint. So you have a 4/9 probability of getting a vacation that keeps you away from snow for at least two days. Hopefully you found that fun and useful for the next time that you are some type of strange game show.