Mutually exclusive events and independent events. Two or more events are said to be mutually exclusive if occurence of any one of the events excludes the occurence of others. This can also be called disjoint events. Another way to look at it is that events A and B are mutually exclusive or disjoint if they can't occur simultaneously. This concept is often easier to understand with examples. I'm going to give you four scenarios and I want you to think about whether they would be mutually exclusive or not. First, rolling a die one time and getting a two and getting a three. Those two events are mutually exclusive since they can't occur at the same time. We know that when we roll a die exactly one number will show up. If that number is a two, then it's not possible for that number to be a three at the same time. Therefore, the events are mutually exclusive, or disjoint. Now, what about rolling a die one time, and getting an even number and also getting a two. Those two events could happen in the same roll, so they are not disjoint. If we roll an even number on a die, it could be either 2, 4, or 6. So, it's possible that our roll would be an even number that also happens to be 2. Notice that when we say two events are not mutually exclusive it doesn't mean that they must happen at the same time, just that it's possible they could happen at the same time. - My third example is that you encounter traffic on the way to the airport, and your flight leaves late. These are also not mutually exclusive. In fact, as you're getting stressed out in traffic while you drive to the airport, you're probably hoping that your flight will be late, so you don't miss it. Right? Those two things both could happen related to the same flight. So their not this shoint. The final example is a flight arriving on time today plus that same flight being late today and that same flight being canceled today. Those three events are all mutually exclusive. One flight can only be in one of those three states. It's either on time, late or cancelled. It's not possible for the same flight to be any two or all three. The next statistics term to learn is independence. If occurrence of one event does not change the probability of another event occurring, the two events are said to be independent. Another way to look at it is that events A and B are independent if the probability of B occurring is not affected by A occurring. Let's do some examples. Are flipping a head on one coin toss and flipping a head on the next Independent events? Yes, every coin toss has a 50% chance of being heads. This doesn't change even if you've already flipped heads several times at a row. The probability of flipping heads again on that next toss is still 50%. What about rain at your house and rain at your next door neighbor's house? Those are not independent. If its raining at your house, chances are much higher that it is also raining at your next door neighbors. Are receiving defective raw materials in one palette from supplier A and receiving defective materials in another palette from supplier B independent? Yes, we would likely consider this to be independent events. Since the product is coming from two different suppliers, one of them having defective product should not influence the probability of the other having defective product. There are cases where the root cause could be the same, but in general it would be safe to assume these are independent. Finally, what about receiving defective raw materials in one pallet from Supplier A and receiving defective raw materials. And another pallet from, that same Supplier A sent in the same batch. Now, these two events are not likely to be independent. Chances are, that whatever caused the defects in the first pallet, may have also caused defects in the other pallet from that same batch. It's not a guarantee that there'll be defects in the second pallet, but the probability of defects would be higher based on the defects in the first. In practice, independence is not necessarily clear cut like these raw materials examples. We sometimes need to make assumptions about whether two events are independent or not, and we'll talk about that more as we continue in this specialization.