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# Probability with Venn diagrams

AP.STATS:
VAR‑4 (EU)
,
VAR‑4.E (LO)
,
VAR‑4.E.4 (EK)

## Video transcript

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 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 thirteen different types of cards or sometimes it's called the rank so each each suit has 13 types of cards types of cards you have the ACE then you have the 2 the 3 the 4 the 5 the 6 7 8 9 10 and then you have the Jack the king and the Queen and that is 13 cards so you can have for each suit you can have any of these for 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 out playing cards and actually count them take out the Joker's in count'em 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 you like look there's 13 of these ranks or types and each of those come in four different suits thirteen times four once again you have gotten 52 cards now with that out 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 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 there's four Jack's in that Ecch so it is for over 52 these are both divisible by four four divided by four is 152 divided by four is 13 now let's think about loud now let's think about the probability so I'm you know we're going to start over I'm going to put that jack back in 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 it's suit 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 suit 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 there 13 of the 52 would result in hearts and butt and both of these are divisible by 13 this is the same thing as 1/4 1 in 4 times I will pick it out or I have a 1 in 4 probability of getting a hearts when I go to that when I randomly pick a card from that shuffle 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 it's a jack and and it is a hearts it is a jack and it is a hearts well if you're 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 it probability that we pick the exact card the Jack of Hearts well there's only one one event one card that that meets these that meets this criteria right over here and there's 52 possible cards so there's a 1 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've once again I have a deck of 52 cards I shuffle 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 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 it's we know we know first of all that there are 52 possibilities there are 52 possibilities but how many of those possibilities meet the criteria 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 can imagine it has an area of 52 so this is 52 52 possible outcomes now how many of those outcomes result in a jack so we already learned this one out of 13 of those outcomes result result in a jack so I could draw a little circle here where that area and I'm approximating that represents the probability of a jack so it should be roughly 1 13 or 4 or 50 seconds 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 the probability the probability of the jack it is 4 it is there's 4 possible cards out of the 52 so that is 4 or 50 seconds or 1 out of or one out of 13 113 now what's the probability of getting a hearts well I'll draw another little circle here that represents that 13 out of 52 13 out of these 52 cards represent a heart and actually one of them's represents both a heart and a jack so let me 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 heart so this is the number of hearts number of 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 clear that so the number number of Jack's number of Jack's 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 the number of Jack's and hearts so let me draw a little arrow there it's getting a little cluttered if I should have drawn a little bit bigger number of Jack's and and hearts number of Jack's and hearts and that's an overlap over there so what is the probability of getting a jack or heart so if you think about it the problem the probability is going to be the number of events that meet this 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 it's going to be you could say well look the green circle right there says the number that gives us the Jack and the orange circle tells us that the number that gives us a heart so you might want to say well why don't we add up the 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 if you did four plus 13 what are we saying we're saying that there are four we're saying that there are four jacks and we're saying that there are we are saying that there are 13 hearts but in both of these were in both 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 you want to figure out the total area here you want to figure out this total area 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 the circles combined you would look at the area of this circle you would 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 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 if this is if this area has a this area is B and the intersection where they overlap is C is C they're the combined area is going to be a plus B - where they overlap - 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 one out of that so it's going to be 4 plus 13 minus 1 or this is going to be this is going to be 1650 seconds and both of these things are divisible both of these things are divisible by four so this is going to be the same thing as divide 16 by 4 you get 452 divided by 4 is 13 so this is there's a four thirteenth chance that you get a jack or a hearts