### Video Transcript

On the street, 10 houses have a
cat, C, eight houses have a dog, D, three houses have both, and seven houses have
neither. Now this question comes in three
parts. Let’s look at part one first. Find the total number of houses on
the street. Hence, find the probability that a
house chosen at random has both a cat and a dog. Give your answer to three decimal
places.

Now, a great way to tackle this
question is to draw a Venn diagram. In this case, the universal set for
a Venn diagram is all of the houses on the street. The left-hand circle represents the
houses that have a cat. The right-hand circle represents
houses that have a dog. And the intersection of those two
circles is the houses which have both. Anything which is outside the
circles but inside the rectangle is a house that has neither a cat nor a dog. Now we’re told that 10 houses have
a cat, but they’re gonna be distributed across houses which only have a cat and
houses which have a cat and a dog. Likewise, the eight houses that
have a dog are gonna be distributed between houses that only have a dog and those
that have both a cat and a dog.

So it’s gonna be easiest for us to
start off by looking at the houses that have both a cat and a dog, and there are
three of those. And there are 10 houses that have a
cat and three of those are houses which also have a dog. So that leaves 10 minus three,
that’s seven houses, which only have a cat. And of the eight houses that have a
dog, three of them also have a cat so that leaves eight minus three, that’s five,
which only have a dog. And lastly, we’re also told that
seven houses have neither a cat nor a dog. So that’s seven out here.

So the total number of houses on
the street is made up of the seven houses who just have a cat, the five houses who
just have a dog, the three houses who have both a cat and a dog, and the seven
houses that neither have a cat nor have a dog. And when you sum those, you get
22. Now we’ve got to find the
probability that a house chosen at random has both a cat and a dog. One way of thinking about this
probability question is what proportion of houses on the street have both a cat and
a dog. Well, we saw that three houses have
both a cat and a dog, and there are 22 houses altogether. So the proportion of houses having
both a cat and a dog is three over 22. And if you’re picking the houses at
random, then the probability of picking a house with a cat and a dog is the same as
that proportion. And to three decimal places, that’s
0.316.

Part two of the question asks, find
the probability that a house on the street has either a cat or a dog or both. Give your answer to three decimal
places.

Okay, so assuming the house is
going to be chosen at random. This is just a matter of counting
up the cases in which houses have a cat or a dog or both from our Venn diagram. And the probability we’re looking
for is just the number of houses with a cat or dog or both as a proportion of the
total number of houses in the street. So seven houses just have a cat,
five houses just have a dog, and three houses have both. So that’s 15 houses. And we saw earlier that the total
number of houses was 22. So the probability we’re looking
for is 15 over 22. And correct to three decimal
places, that’s 0.682.

Now part three is a conditional
probability question. If a house on the street has a cat,
find the probability that there is also a dog living there.

So we’ve been given the fact that
the house has a cat living there. Given that fact, what’s the
probability that there’s also a dog living there? Looking at our Venn diagram then,
straightaway, we can disregard all the cases of houses which don’t have cats. So we can think of this question
as, of the houses that have cats, what proportion also have dogs? We’re only looking at these seven
houses and these three houses. That’s a total of 10 houses that
have cats. And of those 10 houses, only these
three also have a dog. Three out of the 10 houses that
have cats also have a dog. So the probability that a house has
got a dog, given that they’ve got a cat, is three-tenths. Now, we gave our other answers as
decimals, so let’s do that here as well. The probability that a house has a
dog given that it’s got a cat is 0.3.