Basics of Probability Unions Intersections and Complements

Let’s look at intersections, unions, and complements, in probability context. We’ll discuss the basic concepts, and then work through examples. The intersection of events and is the event that both and occur. It is typically denoted by intersect with the intersection symbol. But you might also see and or possibly just AB, representing the intersection. We are free to switch the ordering of the intersection of events. intersect (B and A) is the same event as intersect (A and B). Here is Venn diagram representation. The rectangle represents the sample space, the circle on the left represents event and the circle on the right represents event The intersection of and is this green region, which is in both and It is where both and occur. Here’s another visual representation of events and Now let’s suppose we start pushing the circles apart. little further, little further, there we go. Let’s look at this scenario, where and do not overlap. Here, and do not share any common ground. They share no part of the sample space, and they cannot both occur. We might casually say that they don’t have an intersection, but mathematicians prefer to say that their intersection is the empty set (their intersection contains no elements). In this situation, we say that and are mutually exclusive. The intersection of mutually exclusive events contains no sample points, so the probability of their intersection is 0. Mutually exclusive events are sometimes called disjoint events. The union of events and is the event that either or or both occurs. In set notation, the union is denoted by this union symbol, union But you might simply see simply or When we use the term or in probability, we are referring to their union, and using the word “or” in the inclusive sense — or means or or both. We are also free to switch the ordering in the union of events. union (B or A) is the same event as union (A or B). Here’s Venn diagram representation of events and The green region represents the union — everything that is in either or or both. The probability of the union of two events can be found with the addition rule. The probability of the union of and is the sum of the individual probabilities, minus the probability of the intersection. Why is that? Well, let’s take look at the Venn diagram. If we add the probability of to the probability of then we’ve added the probability of this intersection *twice*, because the intersection occurs in both and The probability of the intersection should only be included once, so we need to subtract one of those. And that end result is called the addition rule. Recall that if and are mutually exclusive, then they don’t share any sample points, they don’t share any common ground, and the probability of their intersection is 0. So if and are mutually exclusive events, then the addition rule simplifies to this, where the probability of the union is equal to the sum of the individual probabilities. This is sometimes referred to as the “special” addition rule. The complement of an event is the event that does not occur. little more formally, complement is the set of all sample points in the sample space that are not in The complement of event is often denoted by bar, but you may also see with superscript or prime. All 3 of these are very commonly used to denote the complement, so it’s not bad idea to get comfortable with all of them. Here’s Venn diagram representation of the sample space. The black circle represents event and the green region represents complement. complement is everything in the sample space that is not in and complement are mutually exclusive events, and together they make up the entire sample space. Take note of 3 things: The union of and complement is the entire sample space. And, since and complement are mutually exclusive, and the probability of the sample space is 1, this implies that the sum of the probabilities of and complement is 1. And finally, the probability of complement is 1 minus the probability of We’ll often use this complement rule in probability problems, although sometimes it’s so natural that we won’t even consciously realize we’re using it. Let’s work through simple example to illustrate some of these concepts. Suppose we are about to roll an ordinary six-sided die once, and observe the number on the top face. Here is natural way to define the sample space: the set of the 6 possible outcomes. If it’s ordinary die, and we’re rolling it fairly, then it’s reasonable to think that these 6 sample points are equally likely. That won’t be perfectly true in practice, but it’s reasonable approximation to reality in this type of scenario. And suppose we define the following events: is the odd numbers, is the values greater than 3, and is made up of the numbers 2 and 6. Since the outcomes are equally likely, we know that the probability of is 3/6 or 1/2, the probability of is also 3/6 or 1/2, and the probability of is 2/6, or 1/3. What are the complements of events and complement is made up of everything in the sample space that is not in So the event complement is the even numbers: 2, 4, and 6. What is the probability of complement? complement is made up of 3 numbers (2, 4, and 6), and there are 6 equally likely possibilities in the sample space, so the probability of complement is 3/6. Note that this is equal to 1 minus the probability of event Event is made up of the numbers that are greater than 3, so complement is made up of the numbers that are less than or equal to 3, the sample points 1, 2, and 3. The probability of complement is thus 3/6, and this also equals 1-P(F). is made up of the numbers 2 and 6, so complement is made up of the other numbers in the sample space: 1, 3, 4, 5. The probability of complement is thus 4/6, and of course, this equals 1 minus the probability of Now suppose we’re interested in the pairwise intersections. The intersection of and is the set of sample points that are in both and and here that’s just the number 5, that’s the only sample point in the sample space that occurs in both and Since the intersection is made up of 1 out of the 6 equally likely outcomes, the probability of the intersection is 1/6. The intersection of and is the set of sample points that are in both and But there are no elements of the sample space that occur in both, so the intersection between and is the empty set. We sometimes use this symbol to represent the empty set. and are mutually exclusive (they cannot both occur on the same roll), and the probability of their intersection is 0. The and intersection contains just the number 6, and the probability of that intersection is 1 out of 6. Now suppose we’d like to find the union of events and The union of events and is the set of sample points that are in either or or both. 1, 3 and 5 are in and 4, 5, and 6 are in so union is made up of the numbers 1, 3, 4, 5, 6. union is made up of 5 of the 6 equally likely sample points, so the probability of that union is 5 out of 6. Alternatively, to find the probability of their union, we could have used the addition rule: The probability of the union of and is the sum of the individual probabilities, minus the probability of the intersection. and each have probability of occurring of 3 out of 6, and we found on the previous slide that their intersection has probability of 1/6, so by the addition rule the probability of their union is 5/6. This is of course the same as what we found above using arguments based on the sample space and our knowledge of what union is. And if, say, we were interested in the complement of union that would simply be the number 2, as that is the only point in the sample space that is not in or The probability of that complement is 1 out of 6. Suppose that in certain population of adults, 10% have diabetes, 30% high blood pressure (or, little more formally, hypertension), and 7% have both. And suppose that person is randomly selected from this population. Let event represent the event that the randomly selected person has diabetes, and event represent the event that the person has hypertension. If we are randomly selecting from this population, then the probability the randomly selected person has diabetes is .10, since 10% of the population has diabetes. The probability they have hypertension is 0.30, and the probability they have both is 0.07. The complement of event is the event the person does not have diabetes. By the complement rule, that probability the person does not have diabetes is 1 minus 0.1, or 0.90. The complement of event is the event the person does not have hypertension. By the complement rule, the probability the person does not have hypertension is 1 minus 0.30, or 0.7. Now suppose we want to know the probability the person has diabetes or hypertension (or both). That’s the probability of the union of events and (D or H), and we can find that with the addition rule. The probability of union is the probability of plus the probability of minus the probability of the intersection of and All 3 of those probabilities are given here, and in the end we find that the probability that the person has diabetes or hypertension is 0.33. And again that's "or" in the inclusive sense, the probability they have diabetes, or hypertension, or both. It’s often helpful to illustrate the events and the probabilities that relate to them in Venn diagram. Let’s look at how we might do that here. Here are the 4 probabilities that we’ve been given or we’ve worked out so far. And here’s Venn diagram illustrating events and In typical Venn diagram, like the one we are using here, the sizes of the various regions, such as the size of circle don’t have any meaning. The Venn diagram simply illustrates the various regions. So try not to read into the sizes of the regions in typical Venn diagram. Now let’s fill in the various regions and their probabilities of occurring. The event the person has both diabetes and hypertension is represented by intersect and that’s this green region. That probability was given to us as 0.07, so I’m going to put that value here. The event that the person has diabetes but not hypertension is represented by intersect complement, and that’s this green region. Since the entire event has probability of occurring of 0.10, the probability associated with this green region, representing diabetes but not hypertension, must be 0.03. The event that the person has hypertension but not diabetes can be represented by complement intersect Or could flip those two around if wanted to, to be more consistent with the wording, as the ordering of events in an intersection doesn’t matter. And that’s this region in green. The probability of in its entirety is 0.30, so the probability of this green region alone must be 0.23. The event that the person has diabetes or hypertension is the union of and represented by this green region. We’ve already found the probability of the union of those events to be 0.33, using the addition rule. But we could also find it here by adding up the 3 probabilities of the 3 mutually exclusive regions that we see here. 0.03 + 0.07 + 0.23 is 0.33. How about the event that the person has neither diabetes nor hypertension? One way of writing that is by recognizing that this event is the complement of the union. The union is that they have either one or both, so the complement of the union is that they have neither. That’s this green region. And since the probability of the entire sample space is 1, the probability associated with this event must be 0.67. Setting up Venn diagram like this often helps us visualize probability problems and makes them much easier to solve. The concepts of unions, intersections and complements can be extended to more than two events. For example, here’s Venn diagram representing 3 events, and The union of events and is this green region, where or or occurs. The complement of that union is this green region, where neither nor nor occurs. The intersection of and is the event that and and occur, represented by this green region. And the complement of that three-way intersection is this green region, where the 3 events do not all occur. Sometimes these combinations of unions, intersections, and complements can be little difficult to think about, so I’ll work through some more complicated problems involving these concepts in another video.