Probability with playing cards and Venn diagrams Probability of compound events. The Addition Rule. Common Core Standard 457 S-CP.7
Probability with playing cards and Venn diagrams
- Let's do a little bit of probability with playing cards
- 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 thirteen different
- types of cards or sometimes it's called the rank.
- So each suit has thirteen types of cards
- 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 thirteen 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 can 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 and actually
- count them, take up the jokers and count them.
- But if you just multiply this, you have four suits
- each of those suits has thirteen types
- so you're gonna have 4 times 13 cards
- or you gonna have 52 cards in a standard playing deck.
- Another way you can say it is like: look, there is thirteen
- of these ranks or types
- and each of those come in four different suits,
- 13 times 4 once again you'd have gotten 52 cards.
- Now that out of the way, let's think about the probabilities
- of different events.
- So let's say I shuffled that deck,
- I shuffled it really, really well.
- And then I randomly picked 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 can pick anyone 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 jacks in that deck.
- So it is 4 over 52, these are both divisible by four
- Four divided by four is one
- 52 divided by 4 is 13.
- Now let's think about
- the probability, so I will, you know, we're gonna start over
- I'm gonna put that jack back in, I'm gonna 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 randomly picked a card from
- a shuffled deck and it is a hearts? Its 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 thirteen of them are hearts. For each
- of those suits you have thirteen types so there's thirteen
- hearts in that deck, there thirteen diamonds in that deck
- there thirteen spades in that deck, there thirteen clubs in that deck.
- So the 13 of the 52 would result in hearts.
- And both of those are divisible by 13, this is the same
- thing as one forth. One in four times I'll pick it out
- or I'll have one forth probability of getting a hearts
- when I go to that, when I randomly pick a card
- from that shuffled deck.
- Now let's do some things a little bit more interesting
- or maybe it's a little obvious: what's the probability
- that I pick something that is a jack and it is a hearts?
- Well, if you're reasonably familiar with cards, you'll know
- that there's actually one card that is both jack and a heart
- It is literally jack of hearts.
- So we're saying what is the probability we picked exactly the card
- jack of hearts?
- Well, there's only one event, one card that meets these 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 more interesting.
- What is the probability, you may 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 have a deck of 52 cards, I shuffle it
- randomly pick a card from that deck, what is the probability that
- that card I picked from that deck is jack or a heart?
- So it could be the jack of hearts or it could be the jack of diamonds
- or could be the jack of spades or it could be the queen of hearts
- or it could be 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 first of all that 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
- area of 52. So this is 52 possible outcomes, now how many of
- those outcomes result in a jack?
- So we already learned this, one out of thirteen of those outcomes
- result in a jack. So I can draw a little circle here with that area
- and I'm approximating
- that represents the probability of a jack.
- So that 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 probability of a jack. It is four, there's four possible
- cards out of the fifty two. So that's 4/52 or 1/13.
- Now what's the probability of getting a hearts?
- Well, I'll draw another circle here, that represents that.
- 13 out of 52, 13 out of these 52 cards represent a heart.
- And actually one of them represents both a heart and a jack.
- So I'll actually overlap them and hopefully this will make sense
- in a second.
- So there's actually thirteen cards that are heart.
- So this is the number of hearts.
- And actually let me right that top thing that way as well.
- It makes it a little bit clear, we were actually looking at -- clear that
- -- so 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 I'm going to do that in yellow
- since I did that problem in yellow.
- This right over here is a number of jack and hearts
- so let me draw a little arrow there. It's getting a little cluttered
- I've actually should've drawn it little bit bigger.
- The number of jacks and hearts.
- And that's an overlap over there, so what's 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 this conditions over the total number of events.
- Yeah, we already know the total number of events are 52.
- But how many meet this 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 the number
- that gives us a heart. So you might wanna say
- well, why don't we add up the green and the orange,
- but if you did that you'd be double-counting.
- Cause if you add it up, if just did four plus thirteen
- what are we saying?
- We're saying that there are four jacks and we're saying
- that there thirteen hearts.
- But in both of these we're, in both when we're doing it this way
- in both cases we're counting the jack of hearts.
- We're putting jack of hearts here and we're putting a jack of hearts here
- So we're counting jack of hearts twice, even though there's only
- one card there. So you'd have to subtract out where they're common.
- You'd have to subtract out the item that is both the jack
- and a heart.
- So you'd subtract out a one.
- Another way to think about it is
- You really want to figure out the total area here.
- Let me zoom in. 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 want to figure the total area
- of both of the circles combined. You'll 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, if this is, 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 one out of that.
- So it's going to be four plus thirteen minus one.
- Or this is going to be 16/52. And both of those things are divisible
- by four.
- So this is going to be the same thing as
- if I divided sixteen by four you get four, fifty two divided by four
- is thirteen.
- So this is, there's 4/13 chance that you get a jack or a hearts.
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