Video: Mutually Exclusive Events

In this video, we will learn how to identify mutually exclusive events and find their probabilities.

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Video Transcript

In this video, we will learn how to identify mutually exclusive events and then find their probability. If two or more events are mutually exclusive, they cannot happen at the same time. For example, an animal cannot be a cat and a dog, which means being a dog is mutually exclusive with being a cat. And while being a dog and being a cat are mutually exclusive, liking dogs is not mutually exclusive to liking cats. There might be some people who like dogs, some people who like cats, and some people who like both dogs and cats, which means this category is not mutually exclusive. Because both things can be true at the same time. Because there is no overlap between dogs and cats, we call these categories mutually exclusive. You might also hear it called “disjoint.”

Before we look at finding the probability of mutually exclusive events, let’s quickly remind ourselves of some rules of probability. For any event A, if the probability of A is the probability of A occurring, the following is true. The probability of A must be between zero and one. The sum of the probabilities of all possible outcomes is equal to one, that is, 100 percent. The compliment of event A written as A with a bar over it refers to everything that is not A. And the probability of the compliment of A is equal to one minus the probability of A. You might occasionally see the compliment be written as A prime, A with a dash, as well.

But now we want to look at finding the probability of mutually exclusive events and non-mutually exclusive events. Let’s say we want to find the probability that event A or event B happens. We write that, the probability of A or B, mathematically like this. On the left, we have mutually exclusive events. And we want to know the probability of A or B happening. And that would be the whole probability in the blue, the probability of A, plus the whole probability of B, the probability of the yellow.

For example, if you had the probability that someone would choose a dog for a pet and the probability that someone would choose a cat for a pet. The probability of A or B would be the probability that they would choose a dog or a cat. And to find that, we would need to add these two probabilities together.

Let’s try the same thing with our non-mutually exclusive event. The probability of event A is everything in the blue circle. If we add the probability of event B to that — that’s the whole yellow circle — the problem is that, in non-mutually exclusive events, A and B share a probability. They share the probability of A and B. That’s that intersection probability.

To find the probability of non-mutually exclusive events, we take the probability of event A plus the probability of event B. And then we subtract the overlap between them.

Going back to our example from earlier, if we let A be the probability that someone likes dogs, B be the probability that someone likes cats, then the probability of A and B is the people who like both. And that means the probability that people like cats or dogs is equal to the probability that people like dogs includes the people who like both cats and dogs. And the probability that someone likes cats includes the people who like both cats and dogs. And that means we’ve added this group twice.

So if we subtract the people who like both cats and dogs, it cancels out that repeat intersection. So that we have the people that like dogs — and that includes the people that like cats and dogs — plus the people that like only cats. In mutually exclusive events, we don’t have to do this because the probability of A and B equals zero. There’s no overlap. So to find the probability of A or B in mutually exclusive events, we add the probability of A to the probability of B because the probability of A and B is zero. Now we’re ready to look at some examples.

In an animal rescue shelter, 39 percent of the current inhabitants are cats, C, and 41 percent are dogs, D. Find the probability that an animal chosen at random is either a cat or a dog. Find the probability that an animal chosen at random is neither a cat nor a dog.

Let’s see what we know. In an animal rescue shelter, 39 percent of the animals are cats and 41 percent of the animals are dogs. For problem one, we wanna find the probability that an animal chosen at random is either a cat or a dog.

We know that animals can either be a cat or a dog, but not both, which makes these mutually exclusive events. And that means, to find the probability of event C or D happening, we add the probability of event C and the probability of event D. Remember, the probability of event A occurring is the number of ways A can occur over all possible outcomes. Probability of C would be the probability a cat is chosen out of all possible animals.

If 39 percent of all the animals are cats, then there is a probability of 0.39 that a cat would be chosen out of all possible animals. In the same way, if 41 percent of all the animals are dogs, then the probability of choosing a dog out of all possible animals is 0.41. If we combine those two probabilities, 0.39 plus 0.41, we see that the probability of selecting a cat or a dog, if chosen randomly, is 0.80. That also means we know that 80 percent of the animals in the shelter are either a cat or a dog. If 80 percent of the animals are either cats or dogs, then 20 percent of the animals are not cats or dogs.

To find the probability that an animal chosen at random is neither a cat nor a dog, we could find the compliment of the probability that it is a cat or a dog. Since the compliment is everything that is not C or D. And we find that by taking the probability of C or D and subtracting that from one. The probability that you will not select a cat or a dog is then 0.20.

Let’s look at another example. This time, we only need to decide if events are mutually exclusive or not.

Amelia has a deck of 52 cards. She randomly selects one card and considers the following events. Event A, picking a card that is a heart. Event B, picking a card that is black. Event C, picking a card that is not a spade. Are events A and B mutually exclusive? Are events A and C mutually exclusive? Are events B and C mutually exclusive?

Let’s take each one of these questions in turn, starting with the first one. Are events A and B mutually exclusive? Event A is picking a card that is a heart, and event B is picking a card that is black. If we consider a standard deck of 52 cards, event A would be selecting any one of the cards that is hearts. And this is event B, picking a card that is black.

In mutually exclusive events, the probability of A and B is zero. It’s not possible for both events to happen at the same time. When we’re asking, “Are A and B mutually exclusive?,” we should ask, can A and B happen at the same time? Can Amelia choose a card that is a heart and black? No, that is not possible. Since it’s not possible for A and B to be true at the same time, these events are mutually exclusive.

What about events A and C? Event A is the same. Because event C involves picking a card that is not a spade, it could be clubs, hearts, or diamonds. We ask the same question. Can event A and event C happen at the same time? That is possible. Since they can both be true at the same time, these events are not mutually exclusive.

What about B and C? With the same question, can B and C happen at the same time? It is possible to choose a card that is black and is not a spade. Those would be any cards that are clubs. Because event B and C can happen at the same time, they’re not mutually exclusive.

Let’s look at another example.

A small choir has a tenor singer, three soprano singers, a baritone singer, and a mezzo-soprano singer. If one of their names was randomly chosen, determine the probability that it was the name of the tenor singer or the soprano singer.

The choir has one tenor, three sopranos, one baritone, and one mezzo-soprano. If we let them be mutually exclusive, that means the tenor would not also be the baritone. We would say there is six total people. We want to know the probability of tenor or soprano. And because they’re mutually exclusive, we can just add these values together, the probability of a tenor and the probability of a soprano.

Since there’s only one tenor, the probability of randomly selecting that person is one out of six. And since there are three sopranos, the probability of randomly selecting their name would be three out of six. Together, that’s a probability of four-sixths, which can be reduced by dividing the numerator and denominator by two. And that means the probability of randomly selecting a tenor or a soprano is two-thirds.

Let’s look at another example.

A bag contains red, blue, and green balls, and one is to be selected without looking. The probability that the chosen ball is red is equal to seven times the probability that the chosen ball is blue. The probability that the chosen ball is blue is the same as the probability that the chosen ball is green. Find the probability that the chosen ball is red or green.

We want to find the probability that we select a red or a green ball when we draw one out. We know that these events are mutually exclusive because the ball could not be red and green at the same time. The probability of the ball being both red and green is zero. And that means the probability that the ball is red or green will be equal to the probability that the ball is red plus the probability that the ball is green.

So now we need to find those values. We know that the probability that the ball is red is equal to seven times the probability that the ball is blue. And the probability that the ball is green is equal to the probability that the ball is blue. We can also say that the probability of red plus the probability of green plus the probability of blue has to equal one.

Since the sum of the probabilities of all possible outcomes is always equal to one. In this equation, if we substitute the probability of blue and for the probability of green, we can substitute seven times the probability of blue and for red. So seven times the probability of blue plus the probability of blue plus the probability of blue has to equal one.

We can combine like terms and say that we have nine times the probability of blue is equal to one. And that means we could divide both sides of this by nine to show the probability of selecting blue is equal to one-ninth. If the probability of selecting blue is one-ninth, then the probability of selecting green is also one-ninth as they are equal. The probability of selecting red is equal to seven times the probability of selecting blue, which means it will be seven-ninths. To find the probability of red and green, then we combine seven-ninths and one-ninth to get eight-ninths.

It’s probably worth noting here that once we found the probability of blue, we could’ve found the probability that it is not blue. Because there’re only red, green, and blue in the bag, the probability that it is not blue will be the same as the probability of it being red or green, which again would be eight-ninths.

In our final example, we’re given the probability of 𝐴 or 𝐵. And then we’re asked to find the probability of 𝐵.

Suppose 𝐴 and 𝐵 are two mutually exclusive events. Given that the probability of 𝐴 or 𝐵 equals 0.93 and the probability of 𝐴 not 𝐵 is equal to 0.39, find the probability of 𝐵.

The first thing we know is the probability of 𝐴 or 𝐵 equals 0.93. But we also know that these are mutually exclusive events, which means the probability of 𝐴 or 𝐵 equals the probability of 𝐴 plus the probability of 𝐵. It also means the probability of 𝐴 and 𝐵 both happening at the same time is zero. We can represent these two mutually exclusive events with two circles that do not overlap.

We’re also given that the probability of 𝐴 minus 𝐵 equals 0.39. This would be the probability of 𝐴 happening and not 𝐵. And that really tells us that the probability of 𝐴 happening is 0.39. We know the probability of 𝐴 plus the probability of 𝐵 equals 0.93. And then we plug in 0.39 for the probability of 𝐴. To solve for 𝐵, we subtract 0.39 from both sides. And we get the probability of 𝐵 is 0.54.

Finally, let’s review our key points. If two events cannot occur at the same time, they are mutually exclusive, which means the probability of 𝐴 or 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵. And it means the probability of 𝐴 and 𝐵 happening at the same time is zero.

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