Welcome to a lesson on counting
where we'll answer three questions involving
a standard deck of playing cards pictured below.
Number one, how many ways can you select a card
which is either red or a face card?
Number two, how many ways can you select a card
which is both red and a face card?
And number three, how many ways can you select two cards
so that the first one is red
and the second one is a face card?
Because all the problems involve red cards and face cards,
let's let set A equal the set of red cards.
And since half of the cards are red
and half of 52 is 26, there are 26 red cards.
And therefore the cardinality of set A is equal to 26.
And let's let B equal the set of face cards.
There are 12 face cards, four Jacks, four Queens
and four Kings.
And therefore the cardinality of set B equals 12.
Next. A union B
is the set of cards that are red or a face card.
And A intersect B is the set of cards that are red
and a face card.
And we'll talk about the cardinality of the union
and intersection in just a moment.
For number one, again, how many ways can you select a card
which is either red or a face card?
Well, this is equal to the number of cards that are red
or a face card,
which is equal to the cardinality of A union B,
meaning the number of cards that are red or a face card.
Well, it's important to recognize here
that set A and set B are not disjoint
because there are some cards that are red and a face card.
And therefore we have to use the formula shown here
to determine the cardinality of A union B.
The cardinality of A union B equals a cardinality of A
plus a cardinality of B,
minus the cardinality of A intersect B,
which means the number of cards that are red
or a face card is equal to the number of red cards,
plus the number of face cards
minus the number of red face cards.
And let's talk about why we have this difference
in the formula.
The cardinality of A or the number of red cards
we know is equal to 26 because there are 26 red cards here.
And the card analogy of B
is equal to the number of face cards,
which is 12 because there are 12 face cards here.
But notice by counting the number of red cards
and the number of face cards,
we counted these six red face cards twice
which is why we have to subtract them out once
by subtracting the cardinality of A intersect B.
Again, the cardinality of A intersect B is equal to
the number of cards that are red and a face card
which are these six cards here.
So by subtracting these six cards out once
since they're counted twice,
this leaves us with a cardinality of A union B
which is 26 plus 12 minus six, which is equal to 32.
If that's a little confusing,
let's look at this using a Venn diagram.
Where the Venn diagram set A is the set of red cards.
And set B is the set of face cards.
Again, we know there are six cards
that are both red and face cards.
These six cards here, which is why we have a six
in the intersection of set A and set B.
And we also know there are 26 total red cards,
20 of which are red, but not face cards,
which are these 20 cards here.
They're red, but not face cards.
And there are 12 face cards, but six of them are not red.
These six black face cards here,
which is why we have six cards
that are face cards, but not red.
So again, there's a couple of ways we can determine
the cardinality of A union B.
Since we have the Venn diagram, we can simply
highlight the union of set A and set B, which is here,
which does give us 20 plus six plus six, which is 32.
But let's also see why there's a difference in the formula.
The cardinality of A union B
is equal to the cardinality of A
plus the cardinality of B.
Again, notice how we counted the six red face cards twice
which is why we also have to subtract
the cardinality of A intersect B once.
So hopefully this helps explain why
we have this formula for the cardinality of A union B when
sets A and B are not disjoint
or the events are not mutually exclusive.
And now let's take a look at number two,
which we probably already have the answer for.
Number two is, how many ways can you select a card
which is both red and a face card?
Well, this is equal to the number of cards that are
red and a face card,
which again, we already know is equal to six.
The cardinality of A intersect B
equals a number of red face cards,
which we can easily see below is equal to six,
these six cards here.
If we go back to the Venn diagram,
these six cards here that are both red and face cards
does give us the cardinality of A intersect B,
which is a number of ways you can select
one card that is both red and a face card.
And now's lets talk about number three,
which is different because we are selecting two cards.
For number three. How many ways can you select two cards
so the first one is red and the second one is a face card?
So this one's much more involved
because we're selecting two cards
and there are also two cases to consider.
Well, let's call case one
the first red card is not a face card
and the second card is a face card.
And case two is the first red card is a face card
and then the second card is a face card.
So because we have an or
and we're selecting two cards, we're going to have a sum of
two products.
Lets begin by considering case one where the first red card
is not a face card.
So how many ways can we select a red card
that is not a face card?
And that would be 20
because there are 20 cards here that are red
and not a face card.
And then times the number of ways the second card
is a face card.
Well, notice how there are a total of 12 face cards.
These 12 cards here.
And therefore we have times 12.
So the number of ways
the first card is red and not a face card.
And the second card is a face card is 20 times 12.
And then we have or the second case
where the first red card is a face card
and the second card is a face card.
So how many ways can we select the first card
that is red and a face card?
Well, there are a total of six red face cards.
So there are six ways to select the first card
that is red and a face card.
And just for the sake of it, let's say the first
red face card was this red face card here.
So it's been picked.
And then we have
the number of ways the second card can be a face card.
Well, notice now there's only 11 face cards left,
these 11 cards here.
And therefore we have six times 11
for the number of ways the first red card is a face card
and the second card is a face card.
So this gives us 20 times 12 plus six times 11,
which is equal to 306.
There are 306 ways to select two cards
so that the first one is red
and the second one is a face card.
I hope you found this helpful.