# Probability with playing cards and Venn diagrams | Probability and Statistics | Khan Academy

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.

Great video as usual….

@kunairuto sal's just that good

the fat guy from the hangover 1 watched this

@kunairuto If you think you know everything about something, then you clearly don't know all about it.

@antixone1 But the cards are not known to the players still in the hand. Hope i understood your point correctly…

Thanks Sal *goes to the casino*

great vid..

@kunairuto gg

"I know almost everythng about math" Humble ain't he? I bet he will be amazed when he finds out he knows almost nothing about it. There is not a single person in the world who knows "almost everything" about any topic. We must not forget how small we are compared with Nature. How can we even say we know a little about any topic? Nature is so complex, yet humans tend to think they are god. Rethink about your place in the universe.

@Shoyrou – My math prof said that Poincare was probably the last mathematician who had mastered all the known mathematics of his day.

Thank you, Mr. Khan, for providing the Common Core Standards. I know it is an extra step for you, but this teacher appreciates it. (Yes, as a teacher, I enjoy your videos. Sometimes, it makes me rethink how I want to teach the same subject to reach my students.)

counting cards 101 🙂

at 1:00 he draws king before queen

@kunairuto "I know almost everything about math"

I'm calling bullshit

In your last calculation, 'probability jacks or hearts' I think there should be added something more :

either you include jack of hearts to that : [P (J and/or hearts)]

or you do not include the jack of hearts : [P (J or hearts)], meaning that it can not be both jack and hearts

in the last case there should be substracted 2 in stead of 1 in the ending sum :

P (J or hearts) = (13+4-2) / 52

if i would be making a mistake saying this please notify me on that 🙂

thanks for educating me! 🙂

@NlessJurnE Sal is emphasizing that he is using inclusive "or" rather than an exclusive "or". Notice he explains how you count that middle section both if you just add the jacks and the hearts.

know more about probability

Can you just fucking tell me when the probability of getting a king or something. FUCK

this video is nice enough. however, why are khanacademy videos coming up as the highest rated no matter what probability/statistics videos I search for. This is impossible. Either khanacademy is telling people to vote for them, or google/youtube is weighting in favor of khanacademy videos.

because , YouTube always provide the best pal !!..

its just me

area 52, I lol'd

yawn

Do a video about poker strategy and probabilities or perhaps card counting in blackjack

8:18 :)))))

Math, not even once.

why dont u?

if ur tired, why did u come here??

Two cards are drawn from a regular deck of 52 cards, without replacement. What is the probability that the first card is an ace of clubs and the second is black?

haha same here

Amazing. WOOOOOOW, Best one I saw explaining "Probability with Playing Cards"

Thanks

because you enjoy shouting out the answers?

Thank u so much Khan Academy, u have no idea how much you've helped me man.

What happens if we have a Joker? :/

What program do you use to draw the maths stuff on?????

Probe-a-booty !?

these lectures are great. Thank you very much for taking your time to do every one of them..Also, your handwriting is awesome.

u keep puting those captions directly on the video……..not seeing the math properly

beautiful explanation! thank you sooo much!

What is the sample space for the number of aces in a hand of 13 playing cards?

y subtract one and not 1/52

It would be of easier understanding if, in Venn's diagram, you put 'three' in the Jack's circle, 'twelve' in the Heart's circle and 'one' in the overlap. Besides that, great video, as always. Greetings.

Very helpful

tussi cha gaye :p

The suit burns better

excellent

u explain a little extra than needed

I'm ugly

@Iva Nachkova

You call batman

J, K, Q . . . . ow, my OCD.

what website do u use

It all makes sense now…

Okay I kind of understand a little bit. I was just sitting in class staring at the board not knowing what the hell my teacher was talking about l.

Nice video

Nice video

thanks so useful.

from cards two cards are drawn at random so what's the probability that one is king and the other is queen

I hate cards

they are so deceitful

I used to be a fan, now I'm a lightbulb.

Probability is one of those things for me that should be super easy but I feel like I'm always doing something wrong. Thanks for the help Sal.

thanks Khan this helped i have a test and im really freaked out but this just made me rest well ready for the test

mad don't know to put video then why are you putting

😡😡😈😈

Thanks

Tnx I learned a lot and it actually helped me with my probability chapter

this is so confusing!!!

my brain is totally

Jackednow….Wow wo wo wo wow..😀

we are not familiar with cards .take another option.

8:16–8:20 khanacademy.exe. has stopped working…………lol

It’s J,Q,K lmao

I have a question. My question says that “Bob selects a card from a standard 52 card deck . George selects from a separate deck. What is the probability that both picks a queen.” Thank you and can you’ll tell me the answer and how to work through it

wrg, no worry or feel good or not about that, doesn't matter, no worry no matter what, do things not worry things

great vid thank you

Thank u

Very nice

8:13 you lost me.

2:31 "Я ложу вольта обратно в колоду…"

came here because I didn't get our teacher

So to substract these two is to subtract the one in common to seperate them

Has an “ahhh” uhuh moment

You are a good teacher xD you make clear sense ^^

The other way to do it

Without the venn diagram

Is

You know you have 4 jacks

13 hearts

And 1 in common

4+13 From (4/52 jacks and 13 /52 hearts)

= 17

-1 that they have in common which is a jack that’s also of hearts

=16

But obliviously the visualizing way you did it is great

When you are starting to get it ^^

Thank you so much I absolutely hate math and chemistry and avoid it like the plague

But I ended up stuck with bio -12

That has some chemistry

And data management

Which has statistics and probability math

Which I need for psychology

Practice for Uni

So videos and teachers like you really will help my heavy angst and anxiety

When I don’t naturally “get it”

Does this apply to heart of the cards?

Thanks a lot of help given by you 👍🤔🤗🤫🤪😎😎😎🤓🤓🤓🤓🤓🤓🤓🤓🤓

I Don’t get cards so doing probability with them so annoying

thx

thank u

Why is it always I'm reading the comment not paying attention to anything hes saying ? Also It was so interested I actually wrote a comment.

do you have a video dealing with Yahtzee probability?