In this video, I'll introduce important concepts relating to sets, collections of items. These are very useful to understand probability and also to derive calculation rules for probabilities. They are also a bit special because they're used in logic as well as probability calculus. Let's get started. A sample space is the collection of all possible outcomes for a random phenomenon. For example, these four possible outcomes of tossing a coin two times. And an event is a subset of the sample space. For example, the cases where your last coin toss would result in heads. Now there can be two or more events in a sample space that do not share any outcomes. For example, the case where you double toss which result in zero, one, or two heads. These events are said to be disjoint. Another term for it is mutually exclusive. A special pair of disjoint events is the combination of an event and its opposite, the case that this event is not happening. This opposite case is called the complement. Here it could be no heads versus the other three outcomes. You can also have multiple events which together fill up the complete sample space. These events are called collectively or jointly exhaustive. If they don't overlap they would be disjoint collectively exhaustive. The sum of the probabilities associated with disjoint events will be smaller than or equal to one. But the sum of the probabilities associated with collectively exhaustive events will be equal to one. These concepts can be intuitively understood with so-called Venn diagrams, combinations of simple geometric shapes that represent sets or parts of sets. This rectangle depicts the sample space. Inside this space there is the event A and the complement of A is all the rest. Another event, B, inside the sample space and not overlapping with A, is a disjoint event. If you apply this Venn diagram to the double coin toss, could you think of an arrangement of the four different outcomes in the Venn diagram and describe the events? Having a single heads as outcome could be event A, and having two heads could be event B. The complement of A would then contain both tails tails, and heads heads. This could be another example of a Venn diagram for the double coin toss. Try to arrange the four different outcomes in this diagram as well. Here the two events, A and B, do in fact overlap. A could be event of getting heads as second result and B could be the event of having just a single heads. The outcome tails heads falls in both events. The outcome tails tails is also part of the sample space but not part of either of the events. The overlapping part of the two events is called the intersection. This is the shorthand for referring to the intersection of events A and B. Now let's find the probability for an intersection of two events. If two events are disjoint, things are easy. The probability for their intersection is zero. It's namely impossible for an outcome to be part of both events at the same time. If two events are not disjoint, they overlap, things are slightly more complex. Here, we will assume that we are dealing with independent events. This means that the probability for one event in the example, thrown heads second is not influenced by another event, throwing heads just once. For independent events A and B, the probability for their intersection is the product of the separate probabilities. Here, there are two cases in event A. So, the probability that A occurs is two fourth. And the same holds for event B. So, the probability for the intersection results from multiplying the two probabilities, leading to one fourth. Let me summarize what I have explained in this video. Events in the sample space that do not share any outcomes are called disjoint or mutually exclusive. Multiple events that together fill up a sample space are called collectively or jointly exhaustive. If there are just two disjoint and collectively exhaustive events in a sample space, they are each others complement. The sum of the probabilities associated with disjoint events will be smaller than or equal to one. The sum of the probabilities associated with collectively exhaustive events is one. The intersection of events A and B is a subset of both events. It contains outcomes that are part of A, as well as B. The probability of the intersection of independent events A and B is calculated by multiplying the probability of event A with that of B. For disjoint events, this property is zero by definition.