In the Venn diagram below, Ω is a set of all dogs in a show. A is a set of all dogs with long hair. B is a set of all dogs with floppy ears. Three-quarters of the dogs who have long hair have floppy ears. One-third of the dogs who do not have long hair have floppy ears. a) Find the values of 𝑎 and 𝑏.
There are also two other parts of this question. We’ll get to those in a moment. Remember a Venn diagram is a way of representing groups of information using sets. In this case, we have three sets. We have the universal set. That’s represented here using the symbol Ω and it’s all the dogs in the show. Enclosed within this are two sets: A is the set of dogs with long hair and B is the set of dogs with floppy ears. Notice how these two sets overlap. This tells us that there are some dogs who have both long hair and floppy ears.
Now let’s use the information about the numbers in each set. We know that three-quarters of the dogs who have long hair also have floppy ears. The set of dogs with long hair is A. So three-quarters of this set have floppy ears. That’s the overlap. And it means that one-quarter of the dogs who have long hair do not have floppy ears. We know there are 12 such dogs. So one-quarter of this set is represented by 12.
A is the number of dogs in this overlap. It’s three-quarters. So we can work out the value of 𝑎 by multiplying the value of a quarter which is 12 by three. That’s 36. And it tells us there are 36 dogs in the overlap. They have both long hair and floppy ears. This overlap is sometimes called the intersection of the two sets and is written with this symbol that looks a little bit like the letter n. That means A intersection B.
Next, we know that a third of the dogs who do not have long hair have floppy ears. That’s these: it’s the dogs in set B, but not in set A. We can see that there are 48 dogs with neither long hair nor floppy ears. That must be two-thirds. To find the value of 𝑏, which is one-third, we can halve the value of two-thirds. Half of 48 is two. So 𝑏 is 24. 𝑎 is 36 and 𝑏 is 24.
Part b) One dog is going to be chosen at random from all the dogs in the show. Work out the probability that the dog does not have long hair.
Remember to find the probability of an event occurring, we find the number of ways that event can occur and we divide it by the total number of possible outcomes. We’re looking to find the probability that the dog does not have long hair. So we need to find the number of dogs that don’t have long hair and divide that by the total number of dogs in the show.
Remember A is the set of all dogs with long hair. So we’re interested in the numbers outside of the circle A. That’s 24 and 48. There’s a total of 72 dogs who do not have long hair. We need to divide that by the total number of dogs in the show. That’s 12 plus 36 plus 24 plus 48 which is equal to 120.
The probability that a dog chosen at random does not have long hair is therefore 72 over 120. This simplifies to three-fifths. And we could also represent this as a decimal if we wanted. The probability of choosing a dog which does not have long hair is three-fifths or 0.6.
Part c) One of the dogs with floppy ears is chosen at random. Work out the probability that the dog has long hair.
This time, we narrow down the total number of dogs we’re choosing from. We know we’re choosing from the dogs who have floppy ears. That’s set B. And it’s 36 plus 24 which is 60 dogs. Out of these, 36 also have long hair. Remember that’s the overlap. This means the probability of choosing a dog at random with long hair out of those which have floppy ears is 36 over 60. Once again, this simplifies to three-fifths, which we know it’s the same as 0.6.
The probability is three-fifths or 0.6.