 In this video, we are going to look at set notation, Venn diagrams and probability. If you haven't already seen our basic video on set notation and Venn diagrams, check it out here. Firstly, we need to know what the symbols and notations we might encounter mean. A single capital P simply means, find the probability of what follows. Consider A, A is the name of a set, a collection of things considered to be a single object. We can also have A with an apostrophe, A apostrophe. This means that something is not in set A. We also have something called the universal set. This includes everything we are looking at and usually has the symbol epsilon. We can often see it in the top corner of a Venn diagram, denoting that everything is within the diagram border. Starting with the most basic combination of these topics, let's look at the probability of A. This means, find the probability that something is in the set A. In a Venn diagram, it will look like this. The area shaded is set A, so your probability would be from within this A. Now let's look at the probability of not being in A. This means the probability that a quantity is not in set A. On a Venn diagram, that would be from this shaded area. Now imagine that the universal set contains the integers between 1 and 10. And set A contains only square numbers. The notation for this would look like this. Pause the video and see if you can work out where the values should go. Within a Venn diagram, the values would lie as such with the square numbers 1, 4 and 9 within the set A. From this, try and find the probability of not being in A. Pause the video and have a go. The probability of not being in A means not within the set A. There are 7 values outside set A with 10 in the universal set. Therefore, the probability of picking one of these is 7 over 10. We now need to look at probabilities with two sets involved. When we have two sets, we can look at how they are linked. We use this symbol, called intersection, to say something is in both sets we are looking at. We can use this symbol, union, to say that an object is in one set or the other. On a Venn diagram, with the added set B, the probability of A would look like this. Still just set A shaded. For the probability of not being in A, it would look like this, the inverse. To find the probability of A or B, we first need to identify that we are considering all quantities with A. Because of the union, however, both rules must apply, so we must also consider all values in B. The rule of union allows us to consider all quantities which follow at least one rule. The rule for being in set A or being in set B. This is therefore all values within the sets A and B. In contrast, the probability of A and B is found by considering quantities that follow both rules in the probability. In this case, in set A and in set B. The only area which does so is the center portion, so this section would give us our probability. As you will see, unions will always provide a larger possibility space than intersection. Let's try and apply this to a question. A group of people will ask which sport they play, with the results being displayed in this Venn diagram. Find the probability of not being in A and being in B. Pause and have a go. To find this probability, firstly we must consider that we are dealing with intersection, so must follow both rules. Identifying that it cannot be in set A would leave us this area for our possible values. However, because we must also follow the rule that it is in set B, we are left with the following crossover area. Therefore, to find the probability of not being in A and being in B, we have 20 people out of the total 130 people asked for a probability of 20 over 130 or 2 over 13. Have a go at finding the probability of being in A or not being in B. For this Venn diagram, pause the video and try now. To find the probability of our quantity satisfying the information given, we first need to note that we have union, so only need to follow one of the rules at a time. The probability of being in A or not being in B tells us it must be in A, giving us this shaded area. We can now apply the rule that it mustn't be in B. This is the entire area outside of set B. Because we only need to follow one rule however, in contrast to intersection where we only took the overlapping area, with union we take every area that is shaded. Our probability of being in A or not being in B is therefore the 8 shaded values out of 10 in the universal set, giving us a probability of 8 out of 10 or 4 out of 5 simplified. So there you have a guide to finding probabilities using set notation and Venn diagrams. Be careful to remember that with an intersection you have to follow both rules together, but with a union only one rule must apply at a time.