Question Video: Using the Conditional Probability Formula | Nagwa Question Video: Using the Conditional Probability Formula | Nagwa

# Question Video: Using the Conditional Probability Formula Mathematics • Third Year of Secondary School

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On a street, there are 25 houses, of which 12 have a cat and 4 have both a cat and a dog. If a house has a cat, what is the probability that there is also a dog living there? Give your answer to the three decimal places.

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### Video Transcript

On a street, there are 25 houses, of which 12 have a cat and four have both a cat and a dog. If a house has a cat, what is the probability that there is also a dog living there? Give your answer to the three decimal places.

So in order to find what the probability that a house with a cat also has a dog, what we can do is use the conditional probability formula. So what the conditional probability formula tells us is that the probability of 𝐴 given 𝐵 is equal to the probability of 𝐴 intersection 𝐵 or 𝐴 and 𝐵 divided by the probability of 𝐵. So in our scenario, the probability that a dog lives in a house given that a cat also lives there is equal to the probability of a dog and a cat living at the house divided by the probability of a cat living at the house.

So first of all, what we can work out is what the probability of a cat being at house is. Well, if there are 25 houses in total and 12 have a cat, well then therefore the probability that a cat lives at the house is 12 over 25. And the probability of a dog and a cat living at the house is gonna be equal to. Well, we can see that there are four that have both a cat and a dog. And there are still a total of 25 houses. So that’s gonna give us a probability of four over 25.

Okay, great, we now have both bits that we need to use our conditional probability formula. So now let’s get on and use it. Well, if we do, what we’re gonna get is four over 25 divided by 12 over 25 is the probability of a dog being at a house given that there’s a cat there. Well, we know from the rules of dividing fractions that if we’ve got one fraction divided by another fraction, it’s the same as a fraction multiplied by the reciprocal of the second fraction. So we flip the second fraction.

Well, then what we can do is divide both the numerator and denominator by 25 so we can cross cancel. So we’ve got four multiplied by one over one multiplied by 12, which is gonna be four-twelfths. Then if we divide both the numerator and denominator by four, we’re gonna get one-third. Well, we want the answer to three decimal places, not as a fraction. So what we can do is write out what this is going to be. Well, we could use a calculator and do one divided by three. However, we should know that a third is the same as 0.3 recurring. And what this gives us is 0.333, and that’s to three decimal places. So what this tells us is that if a house has a cat, the probability that there’s also a dog living there is 0.333.

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