Example: Combinatorics and probability Probability of getting a set of cards
Example: Combinatorics and probability
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- A card game using 36 unique cards, four suits, diamonds,
- hearts, clubs, and spades, with cards numbered from 1 to
- 9 in each suit.
- So there's four suits.
- Each of them have nine cards, so that gives
- us 36 unique cards.
- A hand is a collection of nine cards, which can be sorted
- however the player chooses.
- So they're essentially telling us that order does not matter.
- What is the probability of getting all four of the 1's?
- So they want to know the probability of getting all
- four of the 1's.
- So all four 1's in my hand of 9.
- Now this is kind of daunting at first. You're like, gee you
- know, I've got nine cards and I'm taking them out of 36 and
- I have to figure out how do I get all of the 1's.
- But if we think about it just very, very, in very simple
- terms, all a probability is saying is, the number of
- events-- or I guess you could say-- the number of ways in
- which this action or this event happens.
- So this is what the definition of the probability is.
- It's going to be the number of ways in which event can happen
- and when we talk about the event, we're talking about
- having all four 1's in my hand.
- That's the event.
- And all of these different ways, that's sometimes called
- the event space.
- But we actually want to count how many ways that, if I get a
- hand of 9 picking from 36, that I can get
- the four 1's in it.
- So it is the number of ways in which my event can happen and
- we want to divide that into all of the possibilities-- or
- maybe I should write it this way-- the total number of
- hands that I can get.
- So the numerator in blue is the number of different hands
- where I have the four 1's and we're dividing the total
- number of hands.
- Now let's figure out the total number of hands first, because
- on some level this might be more intuitive and we've
- actually done this before.
- Now, the total number of hands, we're
- picking nine cards.
- And we're picking them from a set of 36 unique cards.
- And we've done this many, many times.
- Let me write this, total number of hands, or total
- number of possible hands.
- That's equal to-- you can imagine, you have nine cards
- to pick from.
- The first card you pick, it's going to be 1 of 36 cards.
- Then the next one is going to be 1 of 35.
- Then the next one is going to be 1 of 34, 33, 32, 31.
- We're going to do this nine times, one, two, three, four,
- five, six, seven, eight, and nine.
- So that would be the total number of
- hands if order mattered.
- But we know-- and we've gone over this before-- that we
- don't care about the order.
- All we care about are the actual cars that are in there.
- So we're overcounting here.
- We're overcounting for all of the different rearrangements
- that these cards could have. It doesn't matter whether the
- Ace of diamonds is the first card I pick or
- the last card I pick.
- The way I've counted them right now, we are counting
- those as two separate hands.
- But they aren't two separate hands, so
- order doesn't matter.
- So we have to do is, we have to divide this by the number
- of ways you can arrange nine things.
- So you could put nine of the things in the first position,
- then eight in the second, seven in the third, so
- forth and so on.
- It essentially becomes 9 factorial times 2 times 1.
- And we've seen this multiple times.
- This is essentially 36 choose 9.
- This expression right here is the same thing-- just you can
- relate it to the combinatorics formulas that you might be
- familiar with-- this is the same thing as 36 factorial
- over 36 minus 9 factorial-- that's what this orange part
- is over here-- divided by 9 factorial or over 9 factorial.
- What's green is what's green and what is orange is what's
- orange there.
- So that's the total number of hands.
- Now a little bit more of a nuanced thought process is,
- how do we figure out the number of ways in which the
- event can happen, in which we can have all four 1's.
- So let's figure that out.
- So number of ways-- or maybe we should say this-- number of
- hands with four 1's.
- And just as a little bit of a thought experiment, imagine if
- we were only taking four cards, if a hand only had four
- cards in it.
- Well if a hand only had four cards in it, then the number
- of ways to get a hand with four 1's, there'd only be one
- way, one combination.
- You'd just have four 1's.
- That's the only combination with four 1's, if we were only
- picking four cards.
- But here, we're not only picking four cards.
- Four of the cards are going to be 1's.
- One, two, three, four.
- But the other five cards are going to be different.
- So one, two, three, four, five.
- So for the other five cards-- if you imagine this slot--
- considering that of the 36 we would have to pick four of
- them already in order for us to have four 1's.
- Well, we've used up four of them, so there's 32 possible
- cards over in that position of the hand.
- And then there'd be 31 in that position of the hand.
- And then there'd be 30 because every time we're picking a
- card, were using it up.
- And now we only have 30 to pick from.
- Then we only have 29 to pick from.
- And then we have 28 to pick from.
- And just like we did before, we don't care about order.
- We don't care if we pick the 5 of clubs first or whether we
- pick the 5 of clubs last. So we shouldn't double count it.
- So we have to divide by the different number of ways that
- five cards can be arranged.
- So we have to divide this by the different ways that five
- cards can be arranged.
- The first card or the first position can be any one of
- five cards, then four cards, then three cards, then two
- cards, then one cards.
- So the number of hands with four 1's is
- actually just this number.
- You're actually looking at all of the different ways you can
- fill up the remaining cards.
- These four 1's are just going to be four 1's.
- There's only one way to get that if the remaining cards
- that's going to give all of the different combinations of
- having four 1's.
- So this will be a count of all of the different combinations
- because all of the different extra stuff that you have will
- be all of the different hands.
- Now we know the total number of hands with
- four 1's is this number.
- And now we can divide it by the total
- number of possible hands.
- And I didn't multiply them out on purpose so that we can
- cancel things out.
- So let's do that.
- Let's take this and divide by that.
- So let me just copy and paste it.
- Let's take that and let's divide it by that.
- But dividing by a fraction is the same thing as multiplying
- by the reciprocal.
- So let's just multiply by the reciprocal.
- So let's multiply-- so this is the denominator.
- Let's make this the numerator.
- So let me copy it and then let me paste it.
- So that's the numerator and then that's the
- denominator up there.
- Because we're dividing by that expression.
- So let me-- whoops.
- Let me put that there.
- Let me get the select tool and then let me make sure I'm
- selecting all of the numbers.
- Let me copy it and then let me paste that.
- It's a little messy with those lines there, but I think
- this'll suit our purposes.
- This'll suit our purposes just fine.
- So when we're multiplying by this, we're essentially
- dividing by this expression up here.
- Now this we can simplify pretty easily.
- We have a-- well actually I forgot to do-- this should be
- 9 factorial.
- This should be 9 times 8 times 7 times 6 times 5 times 4
- times 3 times 2 times 1.
- Let me put that in both places.
- Actually let me just-- let me clear that both places.
- Don't want to confuse people.
- I'm sorry if that confused you when I wrote it earlier.
- This would be 9 factorial.
- 9 times 8 times 7 times 6 times 5 times 4 times 3
- times 2 times 1.
- Let me copy and paste that now.
- Copy and then you paste.
- It That's that, right there.
- And then we have this in the numerator.
- We have 5 times 4 times 3 times 1 in the denominator.
- So this will cancel out with that part right over there.
- And then we have 32 times 31 times 30 times 29 times 28.
- That is going to cancel with that.
- That and that cancels out.
- So what we're left with is just this part over here.
- Let me rewrite it.
- So we're left with 9 times 8 times 7 times 6 over-- and
- this will just be an exercise in simplifying this
- expression-- 36 times 35 times 34 times 33.
- And let's see, if we divide the numerator and denominator
- by 9, that becomes a 1, this becomes a 4.
- You can divide the numerator and denominator by 4, this
- becomes a 2.
- This becomes a 1.
- You divide numerator and denominator by 7, this becomes
- a 1, this becomes a 5.
- You can divide both by 2 again and then this becomes a 1.
- This becomes a 17.
- And you could divide this and this by 3.
- This becomes a 2 and then this becomes an 11.
- So we're left with, the probability of having all four
- 1's in my hand of 9 that I'm selecting from 36 unique cards
- is equal to-- in the numerator, I'm just left with
- this 2 times 1 times 1 times 1-- so it's equal to 2 over 5
- times 17 times 11.
- And that is-- so drum roll, this was kind of an involved
- problem-- 5 times 17 times 11 is equal to 935.
- So it's equal to 2 over 935.
- So about roughly 2 in a thousand chance or 1 in a
- 500-- roughly speaking, this isn't exact odds-- you have a
- roughly 1 in 500 chance of getting all four of the 1's in
- your hand of 9 when you're selecting
- from 36 unique cards.
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