Venn diagrams and the addition rule
Let's do a little bit of probability with playing cards. And for the sake of this video, we're going to assume that our deck has no jokers in it. You could do the same problems with the joker, you'll just get slightly different numbers. So with that out of the way, let's first just think about how many cards we have in a standard playing deck. So you have four suits, and the suits are the spades, the diamonds, the clubs, and the hearts. You have four suits and then in each of those suits you have 13 different types of cards-- and sometimes it's called the rank. You have the ace, then you have the two, the three, the four, the five, the six, seven, eight, nine, ten, and then you have the Jack, the King, and the Queen. And that is 13 cards. So for each suit you can have any of these-- you can have any of the suits. So you could have a Jack of diamonds, a Jack of clubs, a Jack of spades, or a Jack of hearts. So if you just multiply these two things-- you could take a deck of playing cards, take out the jokers and count them-- but if you just multiply this you have four suits, each of those suits have 13 types. So you're going to have 4 times 13 cards, or you're going to have 52 cards in a standard playing deck. Another way you could have said, look, there's 13 of these ranks, or types, and each of those come in four different suits-- 13 times 4. Once again, you would have gotten 52 cards. Now, with that of the way, let's think about the probabilities of different events. So let's say I shuffle that deck. I shuffle it really, really well and then I randomly pick a card from that deck. And I want to think about what is the probability that I pick a Jack. Well, how many equally likely events are there? Well, I could pick any one of those 52 cards. So there's 52 possibilities for when I pick that card. And how many of those 52 possibilities are Jacks? Well you have the Jack of spades, the Jack of diamonds, the Jack of clubs, and the Jack of hearts. There's four Jacks in that deck. So it is 4 over 52-- these are both divisible by 4-- 4 divided by 4 is 1, 52 divided by 4 is 13. Now, let's think about the probability. So I'll start over. I'm going to put that Jack back and I'm going to reshuffle the deck. So once again, I still have 52 cards. So what's the probability that I get a hearts? What's the probability that I just randomly pick a card from a shuffled deck and it is a heart? Well, once again, there's 52 possible cards I could pick from. 52 possible, equally likely events that we're dealing with. And how many of those have our hearts? Well, essentially 13 of them are hearts. For each of those suits you have 13 types. So there are 13 hearts in that deck. There are 13 diamonds in that deck. There are 13 spades in that deck. There are 13 clubs in that deck. So 13 of the 52 would result in hearts, and both of these are divisible by 13. This is the same thing as 1/4. One in four times I will pick it out, or I have a one in four probability of getting a hearts when I randomly pick a card from that shuffled deck. Now, let's do something that's a little bit more interesting, or maybe it's a little obvious. What's the probability that I pick something that is a Jack-- I'll just write J-- and it is a hearts? Well, if you are reasonably familiar with cards you'll know that there's actually only one card that is both a Jack and a heart. It is literally the Jack of hearts. So we're saying, what is the probability that we pick the exact card, the Jack of hearts? Well, there's only one event, one card, that meets this criteria right over here, and there's 52 possible cards. So there's a one in 52 chance that I pick the Jack of hearts-- something that is both a Jack and it's a heart. Now, let's do something a little bit more interesting. What is the probability-- you might want to pause this and think about this a little bit before I give you the answer. What is the probability of-- so I once again, I have a deck of 52 cards, I shuffled it, randomly pick a card from that deck-- what is the probability that that card that I pick from that deck is a Jack or a heart? So it could be the Jack of hearts, or it could be the Jack of diamonds, or it could be the Jack of spades, or it could be the Queen of hearts, or it could be the two of hearts. So what is the probability of this? And this is a little bit more of an interesting thing, because we know, first of all, that there are 52 possibilities. But how many of those possibilities meet these conditions that it is a Jack or a heart. And to understand that, I'll draw a Venn diagram. Sounds kind of fancy, but nothing fancy here. So imagine that this rectangle I'm drawing here represents all of the outcomes. So if you want, you could imagine it has an area of 52. So this is 52 possible outcomes. Now, how many of those outcomes result in a Jack? So we already learned, one out of 13 of those outcomes result in a Jack. So I could draw a little circle here, where that area-- and I'm approximating-- represents the probability of a Jack. So it should be roughly 1/13, or 4/52, of this area right over here. So I'll just draw it like this. So this right over here is the probability of a Jack. There's four possible cards out of the 52. So that is 4/52, or one out of 13. Now, what's the probability of getting a hearts? Well, I'll draw another little circle here that represents that. 13 out of 52 cards represent a heart. And actually, one of those represents both a heart and a Jack. So I'm actually going to overlap them, and hopefully this will make sense in a second. So there's actually 13 cards that are a heart. So this is the number of hearts. And actually, let me write this top thing that way as well. It makes it a little bit clearer that we're actually looking at the number of Jacks. And of course, this overlap right here is the number of Jacks and hearts-- the number of items out of this 52 that are both a Jack and a heart-- it is in both sets here. It is in this green circle and it is in this orange circle. So this right over here-- let me do that in yellow since I did that problem in yellow-- this right over here is a number of Jacks and hearts. So let me draw a little arrow there. It's getting a little cluttered, maybe I should draw a little bit bigger number. And that's an overlap over there. So what is the probability of getting a Jack or a heart? So if you think about it, the probability is going to be the number of events that meet these conditions, over the total number events. We already know the total number of events are 52. But how many meet these conditions? So it's going to be the number-- you could say, well, look at the green circle right there says the number that gives us a Jack, and the orange circle tells us the number that gives us a heart. So you might want to say, well, why don't we add up the green and the orange, but if you did that, you would be double counting, Because if you add it up-- if you just did four plus 13-- what are we saying? We're saying that there are four Jacks and we're saying that there are 13 hearts. But in both of these, when we do it this way, in both cases we are counting the Jack of hearts. We're putting the Jack of hearts here and we're putting the Jack of hearts here. So we're counting the Jack of hearts twice, even though there's only one card there. So you would have to subtract out where they're common. You would have to subtract out the item that is both a Jack and a heart. So you would subtract out a 1. Another way to think about it is, you really want to figure out the total area here. And let me zoom in-- and I'll generalize it a little bit. So if you have one circle like that, and then you have another overlapping circle like that, and you wanted to figure out the total area of both of these circles combined, you would look at the area of this circle. And then you could add it to the area of this circle. But when you do that, you'll see that when you add the two areas, you're counting this area twice. So in order to only count that area once, you have to subtract that area from the sum. So if this area has A, this area is B, and the intersection where they overlap is C, the combined area is going to be A plus B-- -- minus where they overlap-- minus C. So that's the same thing over here, we're counting all the Jacks, and that includes the Jack of hearts. We're counting all the hearts, and that includes the Jack of hearts. So we counted the Jack of hearts twice, so we have to subtract 1 out of that. This is going to be 4 plus 13 minus 1, or this is going to be 16/52. And both of these things are divisible by 4. So this is going to be the same thing as, divide 16 by 4, you get 4. 52 divided by 4 is 13. So there's a 4/13 chance that you'd get a Jack or a hearts.