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. Clear. Don't want to confuse people. Clear. 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.