Video: Mutually Exclusive Events

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

17:16

Video Transcript

In this video, we will learn how to identify mutually exclusive events and find their probabilities. Before we begin, you should make sure that you’re already familiar with how to find probabilities of simple events and the basic rules of probability. We’ll start by recapping some of the notation that we’re going to need throughout this video. We’re also going to use Venn diagrams to help illustrate these. But if you aren’t familiar with Venn diagrams, don’t worry. It isn’t essential to this topic.

To describe the probability of a single event 𝐴 occurring, we use the notation 𝑃 of 𝐴, and we may also sometimes see this written with a lowercase 𝑝. We either use the notation 𝐴 bar or 𝐴 prime to represent the complement of the event 𝐴. The complement of an event is all the outcomes in a sample space that are not part of the event itself. So, for example, in the case of rolling a six-sided die, the complement of the event getting a six is all the other outcomes. So that is getting any of the numbers one to five. It’s also important to remember that the sum of the probabilities of an event and its complement is equal to one as, together, the event and its complement cover the entire sample space with no overlap.

If we have two events 𝐴 and 𝐵, then we use this notation here to represent the probability of either event 𝐴 or event 𝐵 occurring or both if that’s possible. This symbol here is the symbol for a union, which you may already be familiar with from set notation. We can read this as the probability of 𝐴 union 𝐵 or you can read it as the probability of 𝐴 or 𝐵. Again, for two events 𝐴 and 𝐵, the probability that both events occur is represented using this notation here. This is called the intersection of two events, so we can read this as either the probability of 𝐴 intersect 𝐵 or the probability of 𝐴 and 𝐵.

Finally, the event 𝐴 minus 𝐵 contains all the elements in the sample space that belong to 𝐴 but do not belong to 𝐵. This is equivalent to the intersection of event 𝐴 and the complement of event 𝐵. Using our die example again, if 𝐴 is the set of outcomes which are prime numbers, so that’s two, three, and five, and 𝐵 is the set of outcomes which are even numbers, so that’s two, four, and six, then 𝐴 minus 𝐵 is the set containing the elements three and five. It’s all the elements that are in set 𝐴 but aren’t in set 𝐵. So, we’ve removed the element two because it’s also in set 𝐵.

We should recall a key rule here that, in general, the probability of 𝐴 minus 𝐵 is equal to the probability of 𝐴 minus the probability of the intersection of 𝐴 and 𝐵. We’ll need to use each of these pieces of notation and these rules throughout our examples. So, if there are any that you’re not already familiar with, take note to them now. Now let’s come onto the new bit, which is this term mutually exclusive. Informally, two events 𝐴 and 𝐵 are said to be mutually exclusive if they cannot happen at the same time or, we could say, if the occurrence of one prevents the occurrence of the other. Essentially, what we mean is that there is no overlap between the two events.

For example, on a die, the events number is less than two and number is greater than four are mutually exclusive as there is no overlap between these two events; there are no numbers on the die or indeed anywhere that is simultaneously less than two and greater than four. More formally, two events 𝐴 and 𝐵 are mutually exclusive if 𝐴 intersect 𝐵 is equal to the empty set 𝜙. This means that the probability of 𝐴 intersect 𝐵 is zero because it’s impossible for both events to occur. If we’re using a Venn diagram to think about mutually exclusive events, then we need to represent them using disjoint circles because, as we’ve seen, there is no intersection between the two events. Let’s consider a simple example of this.

If a die is rolled once, then what is the probability of getting an odd and an even number together?

The events in this question are mutually exclusive as these two events cannot happen simultaneously. There are no numbers which are both odd and even. The intersection of these two events 𝐴 and 𝐵 is the empty set. As there are no elements in the set 𝐴 intersect 𝐵, the probability of this occurring is zero. It’s impossible to have a number which is both odd and even.

So, we’ve seen that for mutually exclusive events, the probability of both events occurring or the probability of their intersection is zero. But what about the probability of one event or the other occurring? Well, let’s think about a die again to see if we can come up with a rule.

We’ll let 𝐴 be the event that the number is odd and 𝐵 be the event that the number is six. These two events are mutually exclusive as six is not an odd number, so they cannot occur at the same time. The list of all possible outcomes in the sample space is the integers from one to six. And as we’re assuming this die is fair, each outcome is equally likely so has a probability of one-sixth. There are three odd numbers on the die, so the probability of event 𝐴 is three-sixths. There’s only one six on the die, so the probability of event 𝐵 is one-sixth.

When we consider 𝐴 union 𝐵, which we can think of as 𝐴 or 𝐵, we’re interested in any outcome which is part of either event 𝐴 or event 𝐵. So that’s any outcome that we’ve circled regardless of the color we’ve used. Four out of six outcomes are circled, so the probability is four-sixths. You may notice that this is equal to the sum of the individual probabilities for events 𝐴 and 𝐵, and that’s no coincidence. This illustrates but doesn’t prove what we call the addition rule for mutually exclusive events. The probability of the union of 𝐴 and 𝐵, which we can also think of as the probability of 𝐴 or 𝐵, is equal to the probability of 𝐴 plus the probability of 𝐵.

We can also visualize this on a Venn diagram if it helps. As the events 𝐴 and 𝐵 are mutually exclusive, we represent them using two disjoint circles. And if we think of each circle as representing a probability, then the total probability of event 𝐴 or event 𝐵 occurring can be found by adding the individual probabilities together. Now it’s important to remember that we can only apply this rule for mutually exclusive events. If the events involved are not mutually exclusive, then we would have overlapping circles on our Venn diagram, and so we’d need a different rule.

Now you may be able to work out how we can tweak the addition rule for non-mutually exclusive events. But that’s beyond the scope of this video. Let’s now consider an example of how we can apply the addition rule for mutually exclusive events.

Two mutually exclusive events 𝐴 and 𝐵 have probabilities the probability of 𝐴 equals one-tenth and the probability of 𝐵 equals one-fifth. Find the probability of 𝐴 union 𝐵.

These two events are mutually exclusive, and so we can recall the addition rule. The probability of 𝐴 union 𝐵 is the sum of the individual probabilities. It’s the probability of 𝐴 plus the probability of 𝐵. For this question then, we have that the probability of 𝐴 union 𝐵 is one-tenth plus one-fifth. We write that fraction of one-fifth as an equivalent fraction with the denominator of 10. It’s two-tenths. And then summing the two probabilities gives three-tenths. By applying the addition rule for mutually exclusive events then, we found that the probability of 𝐴 union 𝐵 is three-tenths.

In our next example, we’ll see how we can apply this addition rule to a slightly more complex problem.

Suppose 𝐴 and 𝐵 are two mutually exclusive events. Given that the probability of 𝐴 bar is 0.61 and the probability of 𝐴 union 𝐵 is 0.76, determine the probability of 𝐵.

Recall, first of all, that 𝐴 bar means the complement of event 𝐴 and that notation for a union means we’re looking at 𝐴 or 𝐵. As these two events are mutually exclusive, we can recall the addition rule. The probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵. We know the probability of 𝐴 union 𝐵. We’re given it in the question; it’s 0.76. And it’s the probability of 𝐵 that we want to find. But in order to do that, we’re going to need to first work out the probability of 𝐴.

We can work this out if we recall that the sum of the probabilities of an event and its complement is always one. We have then that the probability of 𝐴 plus 0.61, that’s the probability given in the question for the probability of 𝐴 complement, is one. And so, the probability of 𝐴 is one minus 0.61, which is 0.39. Substituting back into our addition rule, we have 0.76 is equal to 0.39 plus the probability of 𝐵. So, the probability of 𝐵 is 0.76 minus 7.39, which is 0.37. Remember, we could only apply this addition rule because the events 𝐴 and 𝐵 were mutually exclusive.

In our next problem, we’re going to need an application of one of De Morgan’s laws. This tells us that for any events 𝐴 and 𝐵, which don’t have to be mutually exclusive, the union of 𝐴 complement and 𝐵 complement is equivalent to the complement of the intersection of events 𝐴 and 𝐵. It might be helpful to visualize this one on a Venn diagram. And remember, the events don’t have to be mutually exclusive here. 𝐴 complement is anything which isn’t in circle 𝐴. 𝐵 complement is anything which isn’t in circle 𝐵. The union of these two regions is any area where we’ve put a dot, which we can see is the entire sample space apart from 𝐴 intersect 𝐵. We’ve shaded everything else, so that’s 𝐴 intersect 𝐵 complement.

If 𝐴 and 𝐵 are two mutually exclusive events from a sample space of a random experiment, find the probability of 𝐴 complement union 𝐵 complement.

To answer this question, we need to recall one of De Morgan’s laws: 𝐴 complement union 𝐵 complement is equivalent to the complement of 𝐴 intersect 𝐵. And so, it follows that the probability of 𝐴 complement union 𝐵 complement will be the probability of the complement of 𝐴 intersect 𝐵. Now, this law holds regardless. But these two events in this question are mutually exclusive. And we know that for mutually exclusive events, their intersection is the empty set. This means that the complement of their intersection is the entire sample space, which has a probability of one.

Or more formally, we can say that the probability of 𝐴 intersect 𝐵 complement is one minus the probability of 𝐴 intersect 𝐵. That’s one minus zero, which is one. So, by recalling De Morgan’s law and that for mutually exclusive events the probability of their intersection of zero, we’ve found that for the two mutually exclusive events 𝐴 and 𝐵, the probability of 𝐴 complement union 𝐵 complement is one.

Let’s now consider an example in which we’ll use a different probability rule.

Suppose 𝐴 and 𝐵 are two mutually exclusive events. Given that the probability of 𝐴 minus 𝐵 is 0.52, find the probability of 𝐴.

We recall, first of all, that the event 𝐴 minus 𝐵 contains all the elements in the sample space that belong in set 𝐴 but don’t belong in set 𝐵. It’s equivalent to the intersection of 𝐴 and 𝐵 complement. We also recall a general rule: the probability of 𝐴 minus 𝐵 is equal to the probability of 𝐴 minus the probability of the intersection of 𝐴 and 𝐵. So we have 0.52 is equal to the probability of 𝐴 minus the probability of 𝐴 intersect 𝐵.

But we’re given another key piece of information in the question which we haven’t used yet. These two events 𝐴 and 𝐵 are mutually exclusive. We know that for mutually exclusive events, the probability of their intersection is zero because they cannot occur at the same time. So, the probability of 𝐴 is the same as the probability of 𝐴 minus 𝐵. It’s 0.52.

In our final example, we’ll see how to find probabilities when the sample space is the union of three mutually exclusive events.

Suppose 𝐴, 𝐵, and 𝐶 are three mutually exclusive events in a sample space 𝑆. Given that 𝑆 is the union of 𝐴, 𝐵, and 𝐶, the probability of 𝐴 is one-fifth the probability of 𝐵, and the probability of 𝐶 is four times the probability of 𝐴, find the probability of 𝐵 union 𝐶.

The first key piece of information we’re given is that the sample space is the union of these three mutually exclusive events. This means that these three events entirely partition the sample space with no overlap. And so, the sum of their probabilities is one. We’re also told that the probability of 𝐴 is one-fifth the probability of 𝐵 and the probability of 𝐶 is four times the probability of 𝐴, so it’s four-fifths times the probability of 𝐵. We can therefore form an equation using only the probability of 𝐵. One-fifth the probability of 𝐵 plus the probability of 𝐵 plus four-fifths the probability of 𝐵 is equal to one.

If we simplify, we have one-fifth, one, and four-fifths as coefficients. So, we have that twice the probability of 𝐵 is equal to one. And dividing through by two, we find that the probability of 𝐵 is one-half. Now, the question asked us for the probability of 𝐵 union 𝐶, and so we need to recall the addition rule for mutually exclusive events. This tells us that the probability of their union is equal to the sum of their individual probabilities. We know the probability of 𝐵. It’s one-half, so we just need to calculate the probability of 𝐶.

We’ve already expressed the probability of 𝐶 as four-fifths the probability of 𝐵, so it’s four-fifths multiplied by one-half, which is four-tenths. Using the addition rule for mutually exclusive events then, the probability of 𝐵 union 𝐶 is one-half plus four-tenths. And thinking of one-half as the equivalent fraction five-tenths, we have a total probability of nine-tenths. Remember, we could only apply the addition rule and indeed the fact that the sum of the three probabilities was one because the three events 𝐴, 𝐵, and 𝐶 were mutually exclusive.

Let’s now review the key points that we’ve covered in this video. Two events 𝐴 and 𝐵 are mutually exclusive if they cannot occur at the same time or we could say that there’s no overlap between the two events. More formally, the intersection of the events 𝐴 and 𝐵 is the empty set 𝜙. And so, the probability of the intersection of events 𝐴 and 𝐵 is zero. The addition law for mutually exclusive events tells us that the probability of the union of events 𝐴 and 𝐵 is the sum of their individual probabilities, the probability of 𝐴 plus the probability of 𝐵.

We also found that for mutually exclusive events, the probability of 𝐴 minus 𝐵, that’s the probability of event 𝐴 but not event 𝐵, is just the same as the probability of 𝐴 which is an adaptation of the general rule which tells us that the probability of 𝐴 minus 𝐵 is the probability of 𝐴 minus the probability of the intersection of events 𝐴 and 𝐵. This is because for mutually exclusive events, the probability of their intersection is zero. We can use these rules together with more basic probability rules, such as the rules concerning complements, to answer a wide variety of problems concerning mutually exclusive events.

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