WEBVTT 00:00:00.637 --> 00:00:02.970 Let's do a little bit of probability with playing cards. 00:00:02.970 --> 00:00:04.511 And for the sake of this video, we're 00:00:04.511 --> 00:00:06.864 going to assume that our deck has no jokers in it. 00:00:06.864 --> 00:00:08.780 You could do the same problems with the joker, 00:00:08.780 --> 00:00:11.560 you'll just get slightly different numbers. 00:00:11.560 --> 00:00:13.390 So with that out of the way, let's first 00:00:13.390 --> 00:00:15.800 just think about how many cards we 00:00:15.800 --> 00:00:18.210 have in a standard playing deck. 00:00:18.210 --> 00:00:23.040 So you have four suits, and the suits 00:00:23.040 --> 00:00:26.960 are the spades, the diamonds, the clubs, and the hearts. 00:00:26.960 --> 00:00:29.770 You have four suits and then in each of those suits 00:00:29.770 --> 00:00:31.950 you have 13 different types of cards-- 00:00:31.950 --> 00:00:33.640 and sometimes it's called the rank. 00:00:43.970 --> 00:00:47.920 You have the ace, then you have the two, the three, the four, 00:00:47.920 --> 00:00:52.210 the five, the six, seven, eight, nine, ten, 00:00:52.210 --> 00:00:55.620 and then you have the Jack, the King, and the Queen. 00:00:55.620 --> 00:00:57.590 And that is 13 cards. 00:00:57.590 --> 00:01:00.890 So for each suit you can have any 00:01:00.890 --> 00:01:03.480 of these-- you can have any of the suits. 00:01:03.480 --> 00:01:05.910 So you could have a Jack of diamonds, a Jack of clubs, 00:01:05.910 --> 00:01:09.220 a Jack of spades, or a Jack of hearts. 00:01:09.220 --> 00:01:10.970 So if you just multiply these two things-- 00:01:10.970 --> 00:01:13.532 you could take a deck of playing cards, take out the jokers 00:01:13.532 --> 00:01:15.240 and count them-- but if you just multiply 00:01:15.240 --> 00:01:18.740 this you have four suits, each of those suits have 13 types. 00:01:18.740 --> 00:01:21.120 So you're going to have 4 times 13 cards, 00:01:21.120 --> 00:01:24.207 or you're going to have 52 cards in a standard playing deck. 00:01:24.207 --> 00:01:25.790 Another way you could have said, look, 00:01:25.790 --> 00:01:28.270 there's 13 of these ranks, or types, 00:01:28.270 --> 00:01:31.120 and each of those come in four different suits-- 13 times 4. 00:01:31.120 --> 00:01:33.237 Once again, you would have gotten 52 cards. 00:01:33.237 --> 00:01:34.820 Now, with that of the way, let's think 00:01:34.820 --> 00:01:37.156 about the probabilities of different events. 00:01:37.156 --> 00:01:38.530 So let's say I shuffle that deck. 00:01:38.530 --> 00:01:40.610 I shuffle it really, really well and then 00:01:40.610 --> 00:01:43.240 I randomly pick a card from that deck. 00:01:43.240 --> 00:01:46.400 And I want to think about what is the probability that I 00:01:46.400 --> 00:01:50.220 pick a Jack. 00:01:50.220 --> 00:01:53.450 Well, how many equally likely events are there? 00:01:53.450 --> 00:01:56.540 Well, I could pick any one of those 52 cards. 00:01:56.540 --> 00:02:00.350 So there's 52 possibilities for when I pick that card. 00:02:00.350 --> 00:02:04.130 And how many of those 52 possibilities are Jacks? 00:02:04.130 --> 00:02:07.480 Well you have the Jack of spades, the Jack of diamonds, 00:02:07.480 --> 00:02:09.880 the Jack of clubs, and the Jack of hearts. 00:02:09.880 --> 00:02:14.170 There's four Jacks in that deck. 00:02:14.170 --> 00:02:18.090 So it is 4 over 52-- these are both divisible by 4-- 4 00:02:18.090 --> 00:02:23.060 divided by 4 is 1, 52 divided by 4 is 13. 00:02:23.060 --> 00:02:27.390 Now, let's think about the probability. 00:02:27.390 --> 00:02:28.976 So I'll start over. 00:02:28.976 --> 00:02:30.600 I'm going to put that Jack back and I'm 00:02:30.600 --> 00:02:31.850 going to reshuffle the deck. 00:02:31.850 --> 00:02:34.020 So once again, I still have 52 cards. 00:02:34.020 --> 00:02:37.290 So what's the probability that I get a hearts? 00:02:37.290 --> 00:02:39.290 What's the probability that I just randomly pick 00:02:39.290 --> 00:02:43.720 a card from a shuffled deck and it is a heart? 00:02:43.720 --> 00:02:46.580 Well, once again, there's 52 possible cards 00:02:46.580 --> 00:02:47.480 I could pick from. 00:02:47.480 --> 00:02:51.720 52 possible, equally likely events that we're dealing with. 00:02:51.720 --> 00:02:55.010 And how many of those have our hearts? 00:02:55.010 --> 00:02:57.770 Well, essentially 13 of them are hearts. 00:02:57.770 --> 00:03:00.180 For each of those suits you have 13 types. 00:03:00.180 --> 00:03:01.892 So there are 13 hearts in that deck. 00:03:01.892 --> 00:03:03.350 There are 13 diamonds in that deck. 00:03:03.350 --> 00:03:04.970 There are 13 spades in that deck. 00:03:04.970 --> 00:03:07.490 There are 13 clubs in that deck. 00:03:07.490 --> 00:03:12.570 So 13 of the 52 would result in hearts, and both of these 00:03:12.570 --> 00:03:14.270 are divisible by 13. 00:03:14.270 --> 00:03:16.670 This is the same thing as 1/4. 00:03:16.670 --> 00:03:19.040 One in four times I will pick it out, 00:03:19.040 --> 00:03:21.690 or I have a one in four probability of getting a hearts 00:03:21.690 --> 00:03:24.619 when I randomly pick a card from that shuffled deck. 00:03:24.619 --> 00:03:27.160 Now, let's do something that's a little bit more interesting, 00:03:27.160 --> 00:03:29.250 or maybe it's a little obvious. 00:03:29.250 --> 00:03:32.130 What's the probability that I pick something 00:03:32.130 --> 00:03:38.540 that is a Jack-- I'll just write J-- and it is a hearts? 00:03:42.120 --> 00:03:44.230 Well, if you are reasonably familiar with cards 00:03:44.230 --> 00:03:45.605 you'll know that there's actually 00:03:45.605 --> 00:03:47.900 only one card that is both a Jack and a heart. 00:03:47.900 --> 00:03:49.520 It is literally the Jack of hearts. 00:03:49.520 --> 00:03:50.530 So we're saying, what is the probability 00:03:50.530 --> 00:03:52.870 that we pick the exact card, the Jack of hearts? 00:03:52.870 --> 00:03:56.200 Well, there's only one event, one card, 00:03:56.200 --> 00:04:00.660 that meets this criteria right over here, 00:04:00.660 --> 00:04:02.630 and there's 52 possible cards. 00:04:02.630 --> 00:04:04.630 So there's a one in 52 chance that I 00:04:04.630 --> 00:04:07.650 pick the Jack of hearts-- something that is both a Jack 00:04:07.650 --> 00:04:09.710 and it's a heart. 00:04:09.710 --> 00:04:12.640 Now, let's do something a little bit more interesting. 00:04:12.640 --> 00:04:14.434 What is the probability-- you might 00:04:14.434 --> 00:04:16.600 want to pause this and think about this a little bit 00:04:16.600 --> 00:04:17.808 before I give you the answer. 00:04:17.808 --> 00:04:20.019 What is the probability of-- so I once again, I 00:04:20.019 --> 00:04:22.340 have a deck of 52 cards, I shuffled it, 00:04:22.340 --> 00:04:25.350 randomly pick a card from that deck-- what is the probability 00:04:25.350 --> 00:04:31.420 that that card that I pick from that deck is a Jack or a heart? 00:04:31.420 --> 00:04:33.050 So it could be the Jack of hearts, 00:04:33.050 --> 00:04:35.414 or it could be the Jack of diamonds, 00:04:35.414 --> 00:04:36.830 or it could be the Jack of spades, 00:04:36.830 --> 00:04:38.450 or it could be the Queen of hearts, 00:04:38.450 --> 00:04:39.872 or it could be the two of hearts. 00:04:39.872 --> 00:04:41.330 So what is the probability of this? 00:04:41.330 --> 00:04:43.579 And this is a little bit more of an interesting thing, 00:04:43.579 --> 00:04:46.440 because we know, first of all, that there 00:04:46.440 --> 00:04:49.912 are 52 possibilities. 00:04:49.912 --> 00:04:51.370 But how many of those possibilities 00:04:51.370 --> 00:04:57.010 meet these conditions that it is a Jack or a heart. 00:04:57.010 --> 00:05:00.080 And to understand that, I'll draw a Venn diagram. 00:05:00.080 --> 00:05:02.650 Sounds kind of fancy, but nothing fancy here. 00:05:02.650 --> 00:05:04.740 So imagine that this rectangle I'm drawing here 00:05:04.740 --> 00:05:06.820 represents all of the outcomes. 00:05:06.820 --> 00:05:09.770 So if you want, you could imagine it has an area of 52. 00:05:09.770 --> 00:05:13.730 So this is 52 possible outcomes. 00:05:13.730 --> 00:05:16.730 Now, how many of those outcomes result in a Jack? 00:05:16.730 --> 00:05:20.860 So we already learned, one out of 13 of those outcomes result 00:05:20.860 --> 00:05:21.860 in a Jack. 00:05:21.860 --> 00:05:24.490 So I could draw a little circle here, 00:05:24.490 --> 00:05:27.290 where that area-- and I'm approximating-- represents 00:05:27.290 --> 00:05:28.520 the probability of a Jack. 00:05:28.520 --> 00:05:31.770 So it should be roughly 1/13, or 4/52, 00:05:31.770 --> 00:05:33.460 of this area right over here. 00:05:33.460 --> 00:05:36.060 So I'll just draw it like this. 00:05:36.060 --> 00:05:38.505 So this right over here is the probability of a Jack. 00:05:44.310 --> 00:05:46.660 There's four possible cards out of the 52. 00:05:46.660 --> 00:05:51.560 So that is 4/52, or one out of 13. 00:05:54.270 --> 00:05:56.470 Now, what's the probability of getting a hearts? 00:05:56.470 --> 00:05:58.540 Well, I'll draw another little circle here 00:05:58.540 --> 00:05:59.530 that represents that. 00:05:59.530 --> 00:06:03.540 13 out of 52 cards represent a heart. 00:06:03.540 --> 00:06:05.880 And actually, one of those represents both a heart 00:06:05.880 --> 00:06:06.920 and a Jack. 00:06:06.920 --> 00:06:09.060 So I'm actually going to overlap them, 00:06:09.060 --> 00:06:12.870 and hopefully this will make sense in a second. 00:06:12.870 --> 00:06:18.010 So there's actually 13 cards that are a heart. 00:06:18.010 --> 00:06:19.530 So this is the number of hearts. 00:06:22.180 --> 00:06:25.110 And actually, let me write this top thing that way as well. 00:06:25.110 --> 00:06:27.380 It makes it a little bit clearer that we're actually 00:06:27.380 --> 00:06:32.320 looking at the number of Jacks. 00:06:37.390 --> 00:06:39.210 And of course, this overlap right 00:06:39.210 --> 00:06:42.920 here is the number of Jacks and hearts-- the number 00:06:42.920 --> 00:06:45.470 of items out of this 52 that are both a Jack and a heart-- 00:06:45.470 --> 00:06:47.160 it is in both sets here. 00:06:47.160 --> 00:06:50.810 It is in this green circle and it is in this orange circle. 00:06:50.810 --> 00:06:53.500 So this right over here-- let me do that in yellow 00:06:53.500 --> 00:06:56.600 since I did that problem in yellow-- this right over here 00:06:56.600 --> 00:06:58.210 is a number of Jacks and hearts. 00:06:58.210 --> 00:06:59.710 So let me draw a little arrow there. 00:06:59.710 --> 00:07:01.590 It's getting a little cluttered, maybe 00:07:01.590 --> 00:07:03.325 I should draw a little bit bigger number. 00:07:11.040 --> 00:07:12.960 And that's an overlap over there. 00:07:12.960 --> 00:07:15.670 So what is the probability of getting a Jack or a heart? 00:07:15.670 --> 00:07:18.450 So if you think about it, the probability 00:07:18.450 --> 00:07:20.260 is going to be the number of events 00:07:20.260 --> 00:07:22.977 that meet these conditions, over the total number events. 00:07:22.977 --> 00:07:25.060 We already know the total number of events are 52. 00:07:25.060 --> 00:07:27.040 But how many meet these conditions? 00:07:27.040 --> 00:07:29.950 So it's going to be the number-- you could say, 00:07:29.950 --> 00:07:32.110 well, look at the green circle right there says 00:07:32.110 --> 00:07:35.580 the number that gives us a Jack, and the orange circle tells us 00:07:35.580 --> 00:07:37.580 the number that gives us a heart. 00:07:37.580 --> 00:07:41.100 So you might want to say, well, why don't we add up 00:07:41.100 --> 00:07:43.800 the green and the orange, but if you 00:07:43.800 --> 00:07:46.047 did that, you would be double counting, 00:07:46.047 --> 00:07:47.630 Because if you add it up-- if you just 00:07:47.630 --> 00:07:52.580 did four plus 13-- what are we saying? 00:07:52.580 --> 00:07:55.980 We're saying that there are four Jacks 00:07:55.980 --> 00:08:00.170 and we're saying that there are 13 hearts. 00:08:00.170 --> 00:08:04.230 But in both of these, when we do it this way, in both cases 00:08:04.230 --> 00:08:06.239 we are counting the Jack of hearts. 00:08:06.239 --> 00:08:07.780 We're putting the Jack of hearts here 00:08:07.780 --> 00:08:09.370 and we're putting the Jack of hearts here. 00:08:09.370 --> 00:08:10.869 So we're counting the Jack of hearts 00:08:10.869 --> 00:08:13.890 twice, even though there's only one card there. 00:08:13.890 --> 00:08:17.090 So you would have to subtract out where they're common. 00:08:17.090 --> 00:08:21.090 You would have to subtract out the item that 00:08:21.090 --> 00:08:23.390 is both a Jack and a heart. 00:08:23.390 --> 00:08:24.990 So you would subtract out a 1. 00:08:24.990 --> 00:08:26.800 Another way to think about it is, 00:08:26.800 --> 00:08:29.130 you really want to figure out the total area here. 00:08:33.919 --> 00:08:36.590 And let me zoom in-- and I'll generalize it a little bit. 00:08:36.590 --> 00:08:38.090 So if you have one circle like that, 00:08:38.090 --> 00:08:41.030 and then you have another overlapping circle like that, 00:08:41.030 --> 00:08:43.730 and you wanted to figure out the total area of both 00:08:43.730 --> 00:08:45.700 of these circles combined, you would 00:08:45.700 --> 00:08:47.260 look at the area of this circle. 00:08:50.340 --> 00:08:53.730 And then you could add it to the area of this circle. 00:08:53.730 --> 00:08:56.530 But when you do that, you'll see that when you add the two 00:08:56.530 --> 00:08:59.270 areas, you're counting this area twice. 00:08:59.270 --> 00:09:01.270 So in order to only count that area once, 00:09:01.270 --> 00:09:04.510 you have to subtract that area from the sum. 00:09:04.510 --> 00:09:10.200 So if this area has A, this area is B, 00:09:10.200 --> 00:09:15.830 and the intersection where they overlap is C, 00:09:15.830 --> 00:09:20.370 the combined area is going to be A plus B-- -- 00:09:20.370 --> 00:09:23.492 minus where they overlap-- minus C. 00:09:23.492 --> 00:09:24.950 So that's the same thing over here, 00:09:24.950 --> 00:09:26.874 we're counting all the Jacks, and that 00:09:26.874 --> 00:09:28.040 includes the Jack of hearts. 00:09:28.040 --> 00:09:29.664 We're counting all the hearts, and that 00:09:29.664 --> 00:09:31.230 includes the Jack of hearts. 00:09:31.230 --> 00:09:33.250 So we counted the Jack of hearts twice, 00:09:33.250 --> 00:09:35.260 so we have to subtract 1 out of that. 00:09:35.260 --> 00:09:37.640 This is going to be 4 plus 13 minus 1, 00:09:37.640 --> 00:09:40.284 or this is going to be 16/52. 00:09:42.850 --> 00:09:48.250 And both of these things are divisible by 4. 00:09:48.250 --> 00:09:51.890 So this is going to be the same thing as, divide 16 by 4, 00:09:51.890 --> 00:09:52.670 you get 4. 00:09:52.670 --> 00:09:55.330 52 divided by 4 is 13. 00:09:55.330 --> 00:10:01.460 So there's a 4/13 chance that you'd get a Jack or a hearts.