Lesson Video: Operations on Events Mathematics

In this video, we will learn how to find the probabilities of the complement, intersection, and union of events.

16:10

Video Transcript

In this video, we will learn how to find the probabilities of the complement, intersection, and union of events. We will begin by introducing some key probability notation and definitions together with their representation on a Venn diagram.

The complement of an event 𝐴 in a sample space 𝑆, denoted 𝐴 prime or 𝐴 bar, is the collection of all outcomes in 𝑆 that are not elements of the set 𝐴. In other words, 𝐴 prime is the event that 𝐴 does not occur. The probability rule for complements states that the probability of 𝐴 prime is equal to one minus the probability of 𝐴. The complement of event 𝐴 can be represented on a Venn diagram by the shaded region shown. We shade the entire region outside the circle 𝐴.

We will now recall the definitions for the intersection and union of two events. The intersection of events 𝐴 and 𝐵, denoted 𝐴 intersect 𝐵, is the collection of all outcomes that are elements of both of the sets 𝐴 and 𝐵. In other words, 𝐴 intersect 𝐵 is the event that both 𝐴 and 𝐵 occur. This can be represented on a Venn diagram as shown. We shade the region that is in circle 𝐴 and circle 𝐵. The union of events 𝐴 and 𝐵 denoted 𝐴 union 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. In other words, 𝐴 union 𝐵 is the event that 𝐴 or 𝐵 or both 𝐴 and 𝐵 occur. We can represent the union of events 𝐴 and 𝐵 on a Venn diagram by shading everything inside circle 𝐴 and circle 𝐵.

These two definitions of the intersection and union of events lead us to a key rule of probability. The additive rule of 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. This can also be shown using Venn diagrams. When adding the probability of event 𝐴 to the probability of event 𝐵, we add the intersection twice. This means that by subtracting the probability of the intersection of events 𝐴 and 𝐵 from their sum, we obtain the probability of the union.

The information we have seen so far leads us to six further formulae that we can use to solve problems involving probability on Venn diagrams. Firstly, we have the complement of the union of events 𝐴 and 𝐵. This is equal to one minus the probability of the union of events 𝐴 and 𝐵. On a Venn diagram, this can be shown by shading everything outside of the union of events 𝐴 and 𝐵. In the same way, we have the probability of the complement of the intersection of events 𝐴 and 𝐵. This is equal to one minus the probability of the intersection of events 𝐴 and 𝐵. In this case, we shade everything apart from the intersection of the two events. In the first diagram, we see that the union and its complement sum to one, and in the second diagram, the intersection and its complement sum to one.

Next, we will consider the probability of exactly one event occurring. This leads us to two possible scenarios: firstly, the probability of event 𝐴 occurring and event 𝐵 not occurring and secondly, the probability of event 𝐵 occurring and event 𝐴 not occurring. These are the intersections of event 𝐴 and the complement of event 𝐵 and vice versa. If we want only event 𝐴 to occur, then we shade the region that is in set 𝐴 but not in set 𝐵. This is therefore equal to the probability of 𝐴 minus the probability of 𝐴 intersection 𝐵. The same method can be used in order to demonstrate the probability that just event 𝐵 occurs. This is the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵.

Our last two formulae are the complements of these two. Firstly, we consider the probability of the union of the complement of 𝐴 and event 𝐵, and secondly, the probability of the union of event 𝐴 and the complement of 𝐵. In the Venn diagram drawn, the complement of 𝐴 is shaded in orange and event 𝐵 is shaded in pink. As we want the union of these, one or other or both can occur. This means that we want the entire shaded region. As we have just seen, the area not shaded is the probability that exactly 𝐴 occurs, written the probability of 𝐴 intersection the complement of 𝐵. We can therefore conclude that the probability of the union of the complement of 𝐴 and event 𝐵 is equal to one minus this value. In the same way, we have the probability of the union of 𝐴 and the complement of 𝐵 is equal to one minus the probability of the intersection of the complement of 𝐴 and event 𝐵.

We will now look at a couple of examples where we can use these definitions and formulae to solve problems involving complements, intersections, and unions.

Denote by 𝐴 and 𝐵 two events with probabilities the probability of 𝐴 is 0.2 and the probability of 𝐵 is 0.47. Given the probability of 𝐴 intersection 𝐵 is 0.18, find the probability of 𝐴 union 𝐵.

We begin by recalling the notation for union and intersection in this question. We can then answer this question using the additive rule of probability. This 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. This can also be shown using Venn diagrams. Substituting in the values given, we have the probability of 𝐴 union 𝐵 is equal to 0.2 plus 0.47 minus 0.18. This is equal to 0.49. If the probability of 𝐴 is 0.2, the probability of 𝐵 is 0.47, and the probability of 𝐴 intersection 𝐵 is 0.18, then the probability of 𝐴 union 𝐵 is 0.49.

We could also have worked this out by completing each individual section of the Venn diagram. Clearing some space, we can begin by adding the probability of the intersection of events 𝐴 and 𝐵, which is 0.18. As the probability of event 𝐴 is 0.2, we can calculate the probability of only event 𝐴 occurring by subtracting 0.18 from 0.2. This is equal to 0.02. Likewise, the probability of only event 𝐵 occurring is equal to 0.47 minus 0.18. This is equal to 0.29. The union of the two events will therefore be equal to the sum of 0.02, 0.18, and 0.29. This is equal to 0.49. To complete the Venn diagram, it is important to add the probability that neither event 𝐴 nor event 𝐵 occur, in this case, 0.51. This is one minus 0.49.

In our next example, we need to calculate the probability that neither event 𝐴 nor event 𝐵 occur.

Suppose 𝐴 and 𝐵 are two events with probability 𝑃 of 𝐴 equals 0.6 and 𝑃 of 𝐵 equals 0.5. Given that the probability of 𝐴 intersection 𝐵 is 0.4, what is the probability that neither of the events occur?

In this question, we need to calculate the probability that neither of the events occur. As shown on the Venn diagram, this is the same as the complement of the union of events 𝐴 and 𝐵. As an event and its complement sum to one, we can therefore calculate the probability that neither event 𝐴 nor event 𝐵 occur by subtracting the probability of 𝐴 union 𝐵 from one. In this question, we are not given the probability of the union. However, we are given the probability of event 𝐴, the probability of event 𝐵, and the probability of the intersection of events 𝐴 and 𝐵. This means that we can begin by using the additive rule of probability, which 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.

Substituting in the values given, the right-hand side becomes 0.6 plus 0.5 minus 0.4. The probability of 𝐴 union 𝐵 is equal to 0.7. This is the area shaded in pink on our Venn diagram. We can now find a complement of this by subtracting 0.7 from one, which is equal to 0.3. We can therefore conclude that if the probability of 𝐴 is 0.6, the probability of 𝐵 is 0.5, and the probability of 𝐴 intersection 𝐵 is 0.4, then the probability that neither event 𝐴 nor event 𝐵 occurs is 0.3. This is the section outside of the circles on our Venn diagram.

In our final question, we will use the additive rule of probability to solve a problem in context.

A group of 68 school children completed a survey asking about their television preferences. The results show that 43 of the children watch channel 𝐴, 26 watch channel 𝐵, and 12 watch both channels. If a child is selected randomly from the group, what is the probability that they watch at least one of the two channels?

Our aim in this question is to calculate the probability that a randomly selected child watches at least one of the two channels. This is the probability of 𝐴 union 𝐵 and can be represented on a Venn diagram, as shown. We recall that the additive rule of 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.

We are told in the question that 43 out of the 68 school children watch channel 𝐴. This means that the probability of selecting a child at random that watches channel 𝐴 is 43 out of 68. Likewise, the probability of selecting a child that watches channel 𝐵 is 26 out of 68, as the results of the survey showed that 26 of the children watch channel 𝐵. We were also told that 12 of the children watch both channels. This is the intersection of events 𝐴 and 𝐵 as shown in pink on the Venn diagram. The probability of this intersection is equal to 12 out of 68.

Substituting the three fractions into our formula, we have the probability of 𝐴 union 𝐵 is equal to 43 over 68 plus 26 over 68 minus 12 over 68. As the denominators are the same and 43 plus 26 minus 12 equals 57, the probability of 𝐴 union 𝐵 is 57 over 68. We can therefore conclude that the probability that a randomly selected child watches at least one of the two channels is 57 over 68.

We will now summarize the key points from this video. The complement of an event 𝐴, denoted 𝐴 prime, is the collection of all outcomes that are not elements of set 𝐴. The intersection of events 𝐴 and 𝐵, denoted 𝐴 intersect 𝐵, is the collection of all outcomes that are elements of both of the sets 𝐴 and 𝐵. 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 both of them.

These definitions lead us to the following formulae. The additive rule of 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. Other rules of probability involving complements, unions, and intersections that we have seen in this video include the following six. We can use these rules together with Venn diagrams to calculate probabilities and solve problems in context.

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