0:00:00.637,0:00:02.970
Let's do a little bit of[br]probability with playing cards.

0:00:02.970,0:00:04.511
And for the sake of[br]this video, we're

0:00:04.511,0:00:06.864
going to assume that our[br]deck has no jokers in it.

0:00:06.864,0:00:08.780
You could do the same[br]problems with the joker,

0:00:08.780,0:00:11.560
you'll just get slightly[br]different numbers.

0:00:11.560,0:00:13.390
So with that out of[br]the way, let's first

0:00:13.390,0:00:15.800
just think about[br]how many cards we

0:00:15.800,0:00:18.210
have in a standard playing deck.

0:00:18.210,0:00:23.040
So you have four[br]suits, and the suits

0:00:23.040,0:00:26.960
are the spades, the diamonds,[br]the clubs, and the hearts.

0:00:26.960,0:00:29.770
You have four suits and[br]then in each of those suits

0:00:29.770,0:00:31.950
you have 13 different[br]types of cards--

0:00:31.950,0:00:33.640
and sometimes it's[br]called the rank.

0:00:43.970,0:00:47.920
You have the ace, then you have[br]the two, the three, the four,

0:00:47.920,0:00:52.210
the five, the six,[br]seven, eight, nine, ten,

0:00:52.210,0:00:55.620
and then you have the Jack,[br]the King, and the Queen.

0:00:55.620,0:00:57.590
And that is 13 cards.

0:00:57.590,0:01:00.890
So for each suit[br]you can have any

0:01:00.890,0:01:03.480
of these-- you can[br]have any of the suits.

0:01:03.480,0:01:05.910
So you could have a Jack of[br]diamonds, a Jack of clubs,

0:01:05.910,0:01:09.220
a Jack of spades,[br]or a Jack of hearts.

0:01:09.220,0:01:10.970
So if you just multiply[br]these two things--

0:01:10.970,0:01:13.532
you could take a deck of playing[br]cards, take out the jokers

0:01:13.532,0:01:15.240
and count them-- but[br]if you just multiply

0:01:15.240,0:01:18.740
this you have four suits, each[br]of those suits have 13 types.

0:01:18.740,0:01:21.120
So you're going to[br]have 4 times 13 cards,

0:01:21.120,0:01:24.207
or you're going to have 52 cards[br]in a standard playing deck.

0:01:24.207,0:01:25.790
Another way you could[br]have said, look,

0:01:25.790,0:01:28.270
there's 13 of these[br]ranks, or types,

0:01:28.270,0:01:31.120
and each of those come in four[br]different suits-- 13 times 4.

0:01:31.120,0:01:33.237
Once again, you would[br]have gotten 52 cards.

0:01:33.237,0:01:34.820
Now, with that of[br]the way, let's think

0:01:34.820,0:01:37.156
about the probabilities[br]of different events.

0:01:37.156,0:01:38.530
So let's say I[br]shuffle that deck.

0:01:38.530,0:01:40.610
I shuffle it really,[br]really well and then

0:01:40.610,0:01:43.240
I randomly pick a[br]card from that deck.

0:01:43.240,0:01:46.400
And I want to think about[br]what is the probability that I

0:01:46.400,0:01:50.220
pick a Jack.

0:01:50.220,0:01:53.450
Well, how many equally[br]likely events are there?

0:01:53.450,0:01:56.540
Well, I could pick any[br]one of those 52 cards.

0:01:56.540,0:02:00.350
So there's 52 possibilities[br]for when I pick that card.

0:02:00.350,0:02:04.130
And how many of those 52[br]possibilities are Jacks?

0:02:04.130,0:02:07.480
Well you have the Jack of[br]spades, the Jack of diamonds,

0:02:07.480,0:02:09.880
the Jack of clubs, and[br]the Jack of hearts.

0:02:09.880,0:02:14.170
There's four Jacks in that deck.

0:02:14.170,0:02:18.090
So it is 4 over 52-- these[br]are both divisible by 4-- 4

0:02:18.090,0:02:23.060
divided by 4 is 1, 52[br]divided by 4 is 13.

0:02:23.060,0:02:27.390
Now, let's think[br]about the probability.

0:02:27.390,0:02:28.976
So I'll start over.

0:02:28.976,0:02:30.600
I'm going to put that[br]Jack back and I'm

0:02:30.600,0:02:31.850
going to reshuffle the deck.

0:02:31.850,0:02:34.020
So once again, I[br]still have 52 cards.

0:02:34.020,0:02:37.290
So what's the probability[br]that I get a hearts?

0:02:37.290,0:02:39.290
What's the probability[br]that I just randomly pick

0:02:39.290,0:02:43.720
a card from a shuffled[br]deck and it is a heart?

0:02:43.720,0:02:46.580
Well, once again,[br]there's 52 possible cards

0:02:46.580,0:02:47.480
I could pick from.

0:02:47.480,0:02:51.720
52 possible, equally likely[br]events that we're dealing with.

0:02:51.720,0:02:55.010
And how many of those[br]have our hearts?

0:02:55.010,0:02:57.770
Well, essentially 13[br]of them are hearts.

0:02:57.770,0:03:00.180
For each of those suits[br]you have 13 types.

0:03:00.180,0:03:01.892
So there are 13[br]hearts in that deck.

0:03:01.892,0:03:03.350
There are 13 diamonds[br]in that deck.

0:03:03.350,0:03:04.970
There are 13 spades[br]in that deck.

0:03:04.970,0:03:07.490
There are 13 clubs in that deck.

0:03:07.490,0:03:12.570
So 13 of the 52 would result[br]in hearts, and both of these

0:03:12.570,0:03:14.270
are divisible by 13.

0:03:14.270,0:03:16.670
This is the same thing as 1/4.

0:03:16.670,0:03:19.040
One in four times[br]I will pick it out,

0:03:19.040,0:03:21.690
or I have a one in four[br]probability of getting a hearts

0:03:21.690,0:03:24.619
when I randomly pick a card[br]from that shuffled deck.

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Now, let's do something that's[br]a little bit more interesting,

0:03:27.160,0:03:29.250
or maybe it's a little obvious.

0:03:29.250,0:03:32.130
What's the probability[br]that I pick something

0:03:32.130,0:03:38.540
that is a Jack-- I'll just[br]write J-- and it is a hearts?

0:03:42.120,0:03:44.230
Well, if you are reasonably[br]familiar with cards

0:03:44.230,0:03:45.605
you'll know that[br]there's actually

0:03:45.605,0:03:47.900
only one card that is[br]both a Jack and a heart.

0:03:47.900,0:03:49.520
It is literally[br]the Jack of hearts.

0:03:49.520,0:03:50.530
So we're saying, what[br]is the probability

0:03:50.530,0:03:52.870
that we pick the exact[br]card, the Jack of hearts?

0:03:52.870,0:03:56.200
Well, there's only[br]one event, one card,

0:03:56.200,0:04:00.660
that meets this criteria[br]right over here,

0:04:00.660,0:04:02.630
and there's 52 possible cards.

0:04:02.630,0:04:04.630
So there's a one[br]in 52 chance that I

0:04:04.630,0:04:07.650
pick the Jack of hearts--[br]something that is both a Jack

0:04:07.650,0:04:09.710
and it's a heart.

0:04:09.710,0:04:12.640
Now, let's do something a[br]little bit more interesting.

0:04:12.640,0:04:14.434
What is the[br]probability-- you might

0:04:14.434,0:04:16.600
want to pause this and think[br]about this a little bit

0:04:16.600,0:04:17.808
before I give you the answer.

0:04:17.808,0:04:20.019
What is the probability[br]of-- so I once again, I

0:04:20.019,0:04:22.340
have a deck of 52[br]cards, I shuffled it,

0:04:22.340,0:04:25.350
randomly pick a card from that[br]deck-- what is the probability

0:04:25.350,0:04:31.420
that that card that I pick from[br]that deck is a Jack or a heart?

0:04:31.420,0:04:33.050
So it could be the[br]Jack of hearts,

0:04:33.050,0:04:35.414
or it could be the[br]Jack of diamonds,

0:04:35.414,0:04:36.830
or it could be the[br]Jack of spades,

0:04:36.830,0:04:38.450
or it could be the[br]Queen of hearts,

0:04:38.450,0:04:39.872
or it could be[br]the two of hearts.

0:04:39.872,0:04:41.330
So what is the[br]probability of this?

0:04:41.330,0:04:43.579
And this is a little bit[br]more of an interesting thing,

0:04:43.579,0:04:46.440
because we know, first[br]of all, that there

0:04:46.440,0:04:49.912
are 52 possibilities.

0:04:49.912,0:04:51.370
But how many of[br]those possibilities

0:04:51.370,0:04:57.010
meet these conditions that[br]it is a Jack or a heart.

0:04:57.010,0:05:00.080
And to understand that,[br]I'll draw a Venn diagram.

0:05:00.080,0:05:02.650
Sounds kind of fancy,[br]but nothing fancy here.

0:05:02.650,0:05:04.740
So imagine that this[br]rectangle I'm drawing here

0:05:04.740,0:05:06.820
represents all of the outcomes.

0:05:06.820,0:05:09.770
So if you want, you could[br]imagine it has an area of 52.

0:05:09.770,0:05:13.730
So this is 52 possible outcomes.

0:05:13.730,0:05:16.730
Now, how many of those[br]outcomes result in a Jack?

0:05:16.730,0:05:20.860
So we already learned, one out[br]of 13 of those outcomes result

0:05:20.860,0:05:21.860
in a Jack.

0:05:21.860,0:05:24.490
So I could draw a[br]little circle here,

0:05:24.490,0:05:27.290
where that area-- and I'm[br]approximating-- represents

0:05:27.290,0:05:28.520
the probability of a Jack.

0:05:28.520,0:05:31.770
So it should be[br]roughly 1/13, or 4/52,

0:05:31.770,0:05:33.460
of this area right over here.

0:05:33.460,0:05:36.060
So I'll just draw it like this.

0:05:36.060,0:05:38.505
So this right over here is[br]the probability of a Jack.

0:05:44.310,0:05:46.660
There's four possible[br]cards out of the 52.

0:05:46.660,0:05:51.560
So that is 4/52,[br]or one out of 13.

0:05:54.270,0:05:56.470
Now, what's the probability[br]of getting a hearts?

0:05:56.470,0:05:58.540
Well, I'll draw another[br]little circle here

0:05:58.540,0:05:59.530
that represents that.

0:05:59.530,0:06:03.540
13 out of 52 cards[br]represent a heart.

0:06:03.540,0:06:05.880
And actually, one of those[br]represents both a heart

0:06:05.880,0:06:06.920
and a Jack.

0:06:06.920,0:06:09.060
So I'm actually going[br]to overlap them,

0:06:09.060,0:06:12.870
and hopefully this will[br]make sense in a second.

0:06:12.870,0:06:18.010
So there's actually 13[br]cards that are a heart.

0:06:18.010,0:06:19.530
So this is the number of hearts.

0:06:22.180,0:06:25.110
And actually, let me write this[br]top thing that way as well.

0:06:25.110,0:06:27.380
It makes it a little bit[br]clearer that we're actually

0:06:27.380,0:06:32.320
looking at the number of Jacks.

0:06:37.390,0:06:39.210
And of course,[br]this overlap right

0:06:39.210,0:06:42.920
here is the number of Jacks[br]and hearts-- the number

0:06:42.920,0:06:45.470
of items out of this 52 that[br]are both a Jack and a heart--

0:06:45.470,0:06:47.160
it is in both sets here.

0:06:47.160,0:06:50.810
It is in this green circle and[br]it is in this orange circle.

0:06:50.810,0:06:53.500
So this right over here--[br]let me do that in yellow

0:06:53.500,0:06:56.600
since I did that problem in[br]yellow-- this right over here

0:06:56.600,0:06:58.210
is a number of Jacks and hearts.

0:06:58.210,0:06:59.710
So let me draw a[br]little arrow there.

0:06:59.710,0:07:01.590
It's getting a little[br]cluttered, maybe

0:07:01.590,0:07:03.325
I should draw a little[br]bit bigger number.

0:07:11.040,0:07:12.960
And that's an[br]overlap over there.

0:07:12.960,0:07:15.670
So what is the probability[br]of getting a Jack or a heart?

0:07:15.670,0:07:18.450
So if you think about[br]it, the probability

0:07:18.450,0:07:20.260
is going to be the[br]number of events

0:07:20.260,0:07:22.977
that meet these conditions,[br]over the total number events.

0:07:22.977,0:07:25.060
We already know the total[br]number of events are 52.

0:07:25.060,0:07:27.040
But how many meet[br]these conditions?

0:07:27.040,0:07:29.950
So it's going to be the[br]number-- you could say,

0:07:29.950,0:07:32.110
well, look at the green[br]circle right there says

0:07:32.110,0:07:35.580
the number that gives us a Jack,[br]and the orange circle tells us

0:07:35.580,0:07:37.580
the number that[br]gives us a heart.

0:07:37.580,0:07:41.100
So you might want to say,[br]well, why don't we add up

0:07:41.100,0:07:43.800
the green and the[br]orange, but if you

0:07:43.800,0:07:46.047
did that, you would[br]be double counting,

0:07:46.047,0:07:47.630
Because if you add[br]it up-- if you just

0:07:47.630,0:07:52.580
did four plus 13--[br]what are we saying?

0:07:52.580,0:07:55.980
We're saying that[br]there are four Jacks

0:07:55.980,0:08:00.170
and we're saying that[br]there are 13 hearts.

0:08:00.170,0:08:04.230
But in both of these, when we[br]do it this way, in both cases

0:08:04.230,0:08:06.239
we are counting[br]the Jack of hearts.

0:08:06.239,0:08:07.780
We're putting the[br]Jack of hearts here

0:08:07.780,0:08:09.370
and we're putting the[br]Jack of hearts here.

0:08:09.370,0:08:10.869
So we're counting[br]the Jack of hearts

0:08:10.869,0:08:13.890
twice, even though there's[br]only one card there.

0:08:13.890,0:08:17.090
So you would have to subtract[br]out where they're common.

0:08:17.090,0:08:21.090
You would have to[br]subtract out the item that

0:08:21.090,0:08:23.390
is both a Jack and a heart.

0:08:23.390,0:08:24.990
So you would subtract out a 1.

0:08:24.990,0:08:26.800
Another way to[br]think about it is,

0:08:26.800,0:08:29.130
you really want to figure[br]out the total area here.

0:08:33.919,0:08:36.590
And let me zoom in-- and I'll[br]generalize it a little bit.

0:08:36.590,0:08:38.090
So if you have one[br]circle like that,

0:08:38.090,0:08:41.030
and then you have another[br]overlapping circle like that,

0:08:41.030,0:08:43.730
and you wanted to figure[br]out the total area of both

0:08:43.730,0:08:45.700
of these circles[br]combined, you would

0:08:45.700,0:08:47.260
look at the area of this circle.

0:08:50.340,0:08:53.730
And then you could add it[br]to the area of this circle.

0:08:53.730,0:08:56.530
But when you do that, you'll[br]see that when you add the two

0:08:56.530,0:08:59.270
areas, you're counting[br]this area twice.

0:08:59.270,0:09:01.270
So in order to only[br]count that area once,

0:09:01.270,0:09:04.510
you have to subtract[br]that area from the sum.

0:09:04.510,0:09:10.200
So if this area has[br]A, this area is B,

0:09:10.200,0:09:15.830
and the intersection[br]where they overlap is C,

0:09:15.830,0:09:20.370
the combined area is[br]going to be A plus B-- --

0:09:20.370,0:09:23.492
minus where they[br]overlap-- minus C.

0:09:23.492,0:09:24.950
So that's the same[br]thing over here,

0:09:24.950,0:09:26.874
we're counting all[br]the Jacks, and that

0:09:26.874,0:09:28.040
includes the Jack of hearts.

0:09:28.040,0:09:29.664
We're counting all[br]the hearts, and that

0:09:29.664,0:09:31.230
includes the Jack of hearts.

0:09:31.230,0:09:33.250
So we counted the[br]Jack of hearts twice,

0:09:33.250,0:09:35.260
so we have to subtract[br]1 out of that.

0:09:35.260,0:09:37.640
This is going to be[br]4 plus 13 minus 1,

0:09:37.640,0:09:40.284
or this is going to be 16/52.

0:09:42.850,0:09:48.250
And both of these things[br]are divisible by 4.

0:09:48.250,0:09:51.890
So this is going to be the[br]same thing as, divide 16 by 4,

0:09:51.890,0:09:52.670
you get 4.

0:09:52.670,0:09:55.330
52 divided by 4 is 13.

0:09:55.330,0:10:01.460
So there's a 4/13 chance that[br]you'd get a Jack or a hearts.