Question Video: Solving Probability Problems Using Venn Diagrams

Out of a group of 100 people, 46 have dogs, 41 have cats, and 28 have rabbits. 12 of the people have both dogs and cats, 10 have both cats and rabbits, and 9 have both dogs and rabbits. 8 people have dogs, cats, and rabbits.

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Video Transcript

Out of a group of 100 people, 46 have dogs, 41 have cats, and 28 have rabbits. 12 of the people have both dogs and cats, 10 have both cats and rabbits, and nine have both dogs and rabbits. Eight people have dogs, cats, and rabbits. Find the probability of randomly selecting a person who has dogs, cats, and rabbits. Give your answer as a fraction in its simplest form. Then, find the probability of randomly selecting a person who only has dogs and rabbits. Give your answer as a fraction in simplest form. Third, find the probability of selecting a person who has pets. Give your answer as a fraction in its simplest form. And finally, find the probability of selecting a person who does not have pets. Give your answer as a fraction in its simplest form.

For information like this, one helpful way to organize it is using Venn diagrams. We have three overlapping categories: dogs, cats, and rabbits. When we’re filling out a Venn diagram, especially one that has three categories, it’s helpful to start all the way in the middle, the place where all three of these circles overlap, which will be the people who have all three pets, dogs, cats, and rabbits.

We’re told that eight people have dogs, cats, and rabbits. After this, we’ll want to move out to the sections that overlap between only two circles. Let’s look at the space between dogs and rabbits. We know that nine people have both dogs and rabbits, but because we know that eight people have dogs, cats, and rabbits, this only leaves one person that has dogs and rabbits because we need the entire space between dogs and rabbits to be equal to nine. And we already had eight in the center.

Following the same procedure, we can find out what to put between dogs and cats. 12 people have both dogs and cats, but of those 12, eight of them also have rabbits. That leaves only four people that have dogs and cats without having rabbits. What we’re seeing here is the space between dogs and cats will be equal to 12. What about the group of people who have cats and rabbits but do not have a dog? Well, 10 people have both cats and rabbits, and eight of them have a dog. This means we can say that two of the people who have cats and rabbits do not have a dog.

And now we’re ready to work one final step outward in our Venn diagram. Let’s start with the dogs. If we know that 46 people have dogs, we already have four, eight, and one of those people. So, while 46 people have dogs, some of the people in that category also have other pets. To find the people who only have dogs, we’ll have to say 46 minus four plus eight plus one, which is 33. To find the people who only have cats, we take the 41 people who have cats and subtract the people who have cats and other pets, in our case, four plus eight plus two. And so, we can say that 27 people only have cats.

We now need to find out the number of people who only have rabbits. 28 people have rabbits. And one, eight, and two of them also have other pets. 28 minus one plus eight plus two equals 17. 17 people have rabbits and no other type of pets.

Before we finish, there’s one other group of people we can add to this diagram. We wanna check and see if there’re any people out of our sample set that do not have pets. And if there are, we can place them outside these three circles. To do that we’ll add up the number of people with pets and subtract that from 100. The remaining people will be the people without pets. But we need to be careful that we’re not counting people more than once. And that means we need the people with only dogs, the people with only cats and dogs, the people with cats, dogs, and rabbits, the people with only cats, the people with only cats and rabbits, the people with only rabbits, and the people with only dogs and rabbits.

This Venn diagram that is an intersection of three circles creates seven categories. And that means we’ll add those seven values together. When we do that, we get 92. And if 92 people have pets and 100 people were sampled, eight people do not go in any of the circles and do not have pets.

Maybe you’re thinking, “That seems like a lot of work. Why didn’t we just add up the totals for the dogs, the cats, and the rabbits?” If you try that, 46 plus 41 plus 28 equals 115. This method fails to remove the people who have both cats and dogs, both cats and rabbits, both dogs and rabbits, and all three. If you do that, you end up with duplicates, and that makes your probabilities inaccurate.

Now, we’re ready to use this Venn diagram to answer our four questions. The first probability we’re looking for is selecting a person who has dogs, cats, and rabbits. That’s the very center of our Venn diagram. Eight people have dogs, cats, and rabbits out of the 100 people who were surveyed. Both eight and 100 are divisible by four. And so, in simplest form, the probability of randomly selecting a person who has dogs, cats, and rabbits will be two out of 25.

Next the probability of randomly selecting a person who only has dogs and rabbits. This means the person cannot have a cat. When we look at the intersection of dogs and rabbits, we see two numbers, a one and an eight. But the eight in the middle includes people who have dogs, rabbits, and cats. And that means there is only one person that has dogs and rabbits but doesn’t have cats. The probability of selecting that person is one out of 100.

The probability of selecting a person who has pets would be 92 out of 100. Both 92 and 100 are divisible by four. When we simplify, we then get 23 out of 25.

Finally, the probability of selecting a person who does not have pets are these eight people that did not fit into any category in our Venn diagram, eight out of 100, which simplifies to two out of 25.

One thing it’s worth noting here is that the last two answers will sum to one. The probability of selecting a person who has pets plus the probability of selecting a person who does not have pets should equal one. That will be all of the people in our group. When this happens, we say that these two probabilities are the complement of one another.

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