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Question Video: Conditional Probability Mathematics • 10th Grade

On the street, 10 houses have a cat (C), 8 houses have a dog (D), 3 houses have both, and 7 houses have neither. 1) 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 3 decimal places. 2) Find the probability that a house on the street has either a cat or a dog or both. Give your answer to 3 decimal places. 3) If a house on the street has a cat, find the probability that there is also a dog living there.

04:30

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.

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