Probability using sample spaces
Subsets of sample spaces
- So, this right over here is a screenshot of the Describing Subsets of Sample Spaces exercise on Khan Academy, and I thought I would do a couple of examples, just because it's good practice just thinking about how do we describe sets and subsets. So it reads, Harry Potter is at Ollivanders Wand Shop. As we all know, the wand must choose the wizard, so Harry cannot make the choice himself. He interprets the wand selection as a random process so he can compare the probabilities of different outcomes. The wood types available are holly, elm, maple, and wenge. The core materials on offer are phoenix feather, unicorn hair, dragon scale, raven feather and thestral tail. All right! Based on the sample space of possible outcomes listed below, what is more likely? And so, we see here, we have four different types of woods for the wand, and then each of those could be combined with five different types of core, Phoenix Feather, Unicorn Hair, Dragon Scale, Raven Feather and Thestral Tail. And so, that gives us four different woods and each of those can be combined with five different cores. 20 possible outcomes. And they don't say it here but they way they're talking I guess we can, I'm going to go with the assumption that they're equally likely outcomes, although it would have been nice if they said that, "These are all equally likely" but these are the 20 outcomes. And so, which of these are more likely? The wand that selects Harry will be made of holly or unicorn hair. So, how many of those outcomes involve this? So if, Holly are these five outcomes and if you said, "Holly or Unicorn Hair" it's going to be these five outcomes plus, well this one involves Unicorn Hair but we've already included this one, but the other ones that's not included for the Holly, that involve Unicorn Hair, are the Elm Unicorn, the Maple Unicorn and the Wenge Unicorn. So it's these five, plus these three, right over here. So eight of these 20 outcomes. And if these are all equally likely outcomes, that means there is an 8/20 probability of a wand that will be made of Holly or Unicorn Hair. So this is 8/20 or, that's the same thing as 4/10 or 40% chance. Now, the wand that selects Harry will be made of Holly and Unicorn Hair. Well, Holly and Unicorn Hair, that's only one out of the 20 outcomes. So this, or course, is going to be a higher probability. It actually includes this outcome and then seven other outcomes. So, the first choice includes the outcome for the second choice plus seven other outcomes. So this is definitely going to be a higher probability. Let's do a couple more of these or at least one more of these. You and a friend are playing "Fire-Water-Sponge". I've never played that game. In this game, each of the two players chooses fire, water or sponge. Both players reveal their choice at the same time and the winner is determined based on the choices. I guess this is like Rock, Paper, Scissors. Fire beats sponge by burning it. Sponge beats water by soaking it up. And water beats fire by putting it out. Alright, well, it kind of makes sense. If both players choose the same object, it is a tie. All the possible outcomes of the game are listed below. If we take outcomes one, three, four, five, seven and eight, as a subset of the sample space, which of the statements below describe the subset? So let's look at the outcomes that they have over here. Well, it makes sense that there are nine possible outcomes because, for each of the three choices I can make there's going to be three choices that my friend can make. So, three times three is nine. They've highlighted these red outcomes. Outcome one, three, four, five, seven and eight. So let's see what's common about them. Outcome one, Fire, I get Fire, friend get's Water. OK, so let's see, my friend would win. Outcome three, I pick Fire, my friend does sponge, so actually I would win that one. And then outcome four, Water, Fire. And then outcome five, Water, Sponge. I don't see a pattern just yet,. Let's see what the choice is. The subset consists of all outcomes where your friend does not win. All outcomes where your friend does not win. Well, that's not true because look, outcome one my friend wins. Water puts out fire so, we're not going to select this first choice. So let's see, the subset consists of all the outcomes where your friend wins or there is a tie. So let's see. Where the friend wins or there's a tie. Well, outcome three, this is an outcome where I would win. Or you, or whoever "Your" is, whoever they're talking about. This is one where the friend doesn't win because Fire burns Sponge so, I'm not going to select that one either. Choice three, the subset consists of all of the outcomes where you win or there is a tie. Well, we just said, "Outcome one, I don't win that, my friend wins that. Water puts out the Fire." Now let's look at the last choice. The subset consists of all the outcomes where there is not a tie. Alright, so this is interesting because look, outcome two there is a tie. Outcome six, there is a tie. Outcome nine, there is a tie. There's actually only three scenarios where there's a tie. Either it's Fire, Fire. Water, Water. Or Sponge, Sponge. And those are the ones that are not selected. So, all of these, someone is going to win. Outcome one, three, four, five, seven or eight. So, definitely, definitely go with that one.