Lesson Video: Mutually Exclusive Events Mathematics

Video Transcript

In this video, we will learn how to identify mutually exclusive events and non–mutually exclusive events and find their probabilities. Before we discuss mutually exclusive events, let’s recap compound events and the addition rule of probability.

We recall that the intersection of events 𝐴 and 𝐵 is the collection of all outcomes that are elements of both of the sets 𝐴 and 𝐵. And this is the equivalent to both events occurring. The union of events 𝐴 and 𝐵 is the collection of all outcomes that are elements of one or the other of the sets 𝐴 and 𝐵 or of both of them. This is the equivalent to either of the events occurring.

We also recall that if an event 𝐴 in a sample space 𝑠 cannot occur, then its probability is zero. This means that there are no elements in 𝐴. And we call a set with no elements the empty set. Finally, we recall that the addition rule for probability states that the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵.

Let’s now consider what happens when the probability of this intersection equals zero. If the probability of 𝐴 intersection 𝐵 is equal to zero, then the addition rule for probability would simplify to the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵. This leads us to an informal definition of mutually exclusive events. Two events 𝐴 and 𝐵 where the probability of 𝐴 intersection 𝐵 is equal to zero are mutually exclusive events. This is because both events cannot occur at the same time.

More formally, this can be written as follows. 𝐴 and 𝐵 are mutually exclusive events if 𝐴 intersection 𝐵 is equal to the empty set. This is equivalent to saying the events cannot occur at the same time since the probability of 𝐴 intersection 𝐵 is equal to the probability of the empty set, which we know is equal to zero.

We say that a list of events 𝐴 sub one, 𝐴 sub two, and so on up to 𝐴 sub 𝑛 is mutually exclusive if they are pairwise mutually exclusive. So the intersection of 𝐴 sub 𝑖 and 𝐴 sub 𝑗 is equal to the empty set for any 𝑖 and 𝑗 that exist in the set of numbers one, two, and so on up to 𝑛. Summarizing, if 𝐴 and 𝐵 are mutually exclusive, then the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵. This can be represented on a Venn diagram as shown, where the two circles representing events 𝐴 and 𝐵 do not intersect.

We will now consider some specific examples. And in our first one, we will determine whether the given pairs of events are mutually exclusive.

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; and 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?

In all three parts of this question, we need to determine whether two events are mutually exclusive. We recall that two events 𝑥 and 𝑦 are mutually exclusive if they cannot both occur at the same time, in other words, the probability of 𝑥 intersection 𝑦 is equal to zero.

In this question, we are told that Amelia has a regular deck of 52 cards. We know that these are split into four suits, diamonds, hearts, clubs, and spades, where the first two suits are red cards and the second two are black. Each of the suits has 13 cards: an ace, the numbers two through 10, a jack, a queen, and a king.

There are three events that Amelia needs to consider: firstly, event 𝐴, picking a card that is a heart. This would involve picking any of the 13 cards in the second row. Event 𝐵 involves picking a card that is black. This involves picking a card that is either a club or a spade, any of the 26 cards in the bottom two rows. Finally, we have event 𝐶, which is picking a card that is not a spade. This could be either a diamond, heart, or club, any of the 39 cards in the top three rows.

To determine whether events 𝐴 and 𝐵 are mutually exclusive, we need to find whether there is an outcome that occurs in both events. Is it possible to pick a card that is a heart and pick a card that is black? We know that all hearts are red, so there are no cards which are both black and a heart. And we can therefore conclude that when Amelia is selecting one card, both events cannot happen. And the events are therefore mutually exclusive.

Next, we need to consider whether events 𝐴 and 𝐶 are mutually exclusive. This time, we have the events of picking a card that is a heart and picking a card that is not a spade. The key point here is that all hearts are not spades. This means that choosing any heart will satisfy both events. And since both events can occur at the same time, we can conclude that they are not mutually exclusive. There is an overlap between picking a card that is a heart and picking a card that is not a spade.

Finally, we need to consider whether events 𝐵 and 𝐶 are mutually exclusive. This time, we have the events of picking a card that is black and picking a card that is not a spade. This time, the key fact is that all clubs are black. And they are also not a spade. This means that choosing any of the clubs satisfies both events. And we can therefore conclude that events 𝐵 and 𝐶 are not mutually exclusive. The events 𝐴 and 𝐵 are mutually exclusive, whereas events 𝐴 and 𝐶 and 𝐵 and 𝐶 are not mutually exclusive.

In our next example, we will use the mutual exclusivity of two events and their probabilities to determine the probability of either event occurring.

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.

We begin by recalling that two events 𝐴 and 𝐵 are mutually exclusive if they cannot occur at the same time. This means that there are no elements in event 𝐴 and event 𝐵, and the probability of 𝐴 intersection 𝐵 equals zero. The addition rule of probability tells us that when 𝐴 and 𝐵 are mutually exclusive, the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵. Mutually exclusive events can also be represented on a Venn diagram as shown, where there is no overlap between the circles representing events 𝐴 and 𝐵.

We are told that the probability of event 𝐴 is one-tenth and the probability of event 𝐵 is one-fifth. The probability of 𝐴 union 𝐵 is therefore equal to one-tenth plus one-fifth, which can be rewritten as one-tenth plus two-tenths, which in turn is equal to three-tenths. The probability of 𝐴 union 𝐵 is three-tenths.

In our next example, we will use the mutual exclusivity of three events and properties of probability to determine the probability of a compound event.

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 will begin by naming the events of choosing a red ball, a blue ball, and a green ball as R, B, and G, respectively. Since the selected ball can only be one of these three colors, we can conclude that the events are mutually exclusive. Our aim in this question is to find the probability that the chosen ball is red or green. This is the probability of the union of the two events. And we recall for any two mutually exclusive events 𝑥 and 𝑦, the probability of 𝑥 union 𝑦 is equal to the probability of 𝑥 plus the probability of 𝑦. This means that we need to find the sum of the probability that the chosen ball is red and the probability that the chosen ball is green.

We are told that the probability that the chosen ball is red is seven times the probability that the chosen ball is blue. This can be written as shown. We are also told that the probability that the chosen ball is blue is the same as the probability that the chosen ball is green.

Finally, since there are only red, blue, and green balls in the bag and these are mutually exclusive events, we have the probability of red plus the probability of blue plus the probability of green is equal to one. Replacing the probability of red and the probability of green using equations one and two, we have the following equation. Seven multiplied by the probability of blue plus the probability of blue plus the probability of blue is equal to one. This simplifies to nine multiplied by the probability of blue is equal to one. And the probability that the selected ball is blue is therefore equal to one-ninth.

This means that the probability that the chosen ball is green is also equal to one-ninth. And the probability that the chosen ball is red is seven-ninths. After clearing some space, we can now calculate the probability that the chosen ball is red or green. This is equal to seven-ninths plus one-ninth, which in turn is equal to eight-ninths. When a ball is selected from the bag without looking, the probability that the chosen ball is red or green is eight-ninths.

In our final example, we will see two events that are not mutually exclusive.

The probability that a student passes their physics exam is 0.71. The probability that they pass their mathematics exam is 0.81. The probability that they pass both exams is 0.68. What is the probability that the student only passes their mathematics exam?

We will begin by naming the events of passing examinations in physics and mathematics as 𝑝 and 𝑚, respectively. We are told that the probability of a student passing the physics exam is 0.71. And the probability of passing the mathematics exam is 0.81. Since it is possible to pass both exams, the two events are not mutually exclusive,. And we are told that the probability of passing both exams is 0.68.

This information can be represented on a Venn diagram, where the overlap or intersection of the two circles represents the probability that a student passes both exams. As already mentioned, this is equal to 0.68. The question asked us to calculate the probability that the student only passes their mathematics exam. This will be equal to the probability that they pass mathematics minus the probability that they pass both exams. We need to subtract 0.68 from 0.81. This is equal to 0.13. The probability that the student only passes mathematics, in other words, that they pass mathematics and not physics, is 0.13.

We can add this onto our Venn diagram, as shown. And we could repeat this process to find the probability that the student only passes their physics exam. Subtracting 0.68 from 0.71 gives us 0.03. There is one option left in order to complete the Venn diagram: the probability that the student doesn’t pass physics or mathematics, in other words, that they fail both exams. 0.03 plus 0.68 plus 0.13 is equal to 0.84. Since probabilities must sum to one and one minus this is equal to 0.16, the probability that a student fails physics and mathematics is 0.16.

We now have the completed Venn diagram, which shows that the probability that the student only passes their mathematics exam is 0.13.

We will now summarize the key points from this video. Two events 𝐴 and 𝐵 are mutually exclusive if the intersection of 𝐴 and 𝐵 is equal to the empty set. This means that mutually exclusive events cannot occur at the same time since the probability of 𝐴 intersection 𝐵 is equal to the probability of the empty set, which is equal to zero. A list of events is mutually exclusive if they are pairwise mutually exclusive. Finally, if events 𝐴 and 𝐵 are mutually exclusive, then the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵.