Let's talk a little bit about urns. Everyone who teaches probability is always talking about urns. An urn is a container and you can't see into that container. And there are usually two different colors of marbles, although sometimes there are There are more colors. And the ideas that you're drawing marbles out of the container. There are two basic ways that we can draw marbles out of this container. We can draw them with replacement or without replacement. If we draw them with replacement. then we have a two thirds probability of drawing white marble and a one third probability of drawing a blue marble. Okay. Each time we draw and the draws are independent, so our probability of drawing a white marble two times in a row, two, two. White in a row would be (2/3)(2/3)=4/9. On the other hand, if I am drawing without replacement then on my first draw I may draw a white marble, in which case I have a two thirds probability of doing, in which case my probability of drawing a white marble on the second draw is one half. Or, I could draw a blue marble on the first draw. In which case, there is no chance of drawing two white marbles in a row, okay. So my probability of drawing two white marbles in a row without replacement would be two-thirds times one-half which would be 2/3 times 1/2 would be 2/6 or 3/9. So a minute ago I was telling you that the way that you can arrange people who in different in jobs is kind of a product where you have more slots open at the beginning and fewer at the end. We have a general term for this type of arithmetic where we take a number and we multiply it by a smaller one smaller, the integer one smaller and the integer one smaller than that until we get down to 1. It's called factorial. So this Notation five with an exclamation mark is read "Five Factorial", and it is equal to 5 times 4 times 3 times 2 times 1, which is 120. Okay, when we have. One factorial divided by another. What we're actually saying is that we have this factorial on top, 7.6.5.4.3.2.1 and this factorial on the bottom 5.4.3.2.1 and as you see most of these numbers cancel out, because these Products are equal to 1, and so the resulting number would be 7 times 6. One more thing to know about factorials. By convention, the notation 0 factorial is equal to 1. When we are drawing, N items from a group of m items without replacement. The number of unique groups that we can form has a special name, it's called m choose n. This type of problem comes up so much in probability that it has its own name in its own notation, so if I want to know how many unique committees of five people I could form from a group of ten people. So I have, I start with m equals 10, and n equals 5. This would be described as 10 choose 5. And it can be written using this special notation like this. And what does that equal? It is equal 10 factorial. Divided by 10 minus 5 factorial or 5 factorial times 5 factorial okay? This terminology will come up again and again and again in probability In our current example, we're interested in how many teams, Meaning unique teams where all the rules are equivalent of five people. Can be formed [SOUND] from 10 people. So, we're interested in the number of unique teams. This is referred to as 10 to 5, which is equal to 10 factorial divided by 5 factorial times 5 factorial. And I'm going to show you a little trick for when you want to get an idea of your answer when you don't have your calculator handy. Which is going to be for the 10.9.8.7.6 And then 5 times 4 times 3 times 2 times 1 is going to cancel out the five, one of the five factorials. So, that's cancelled out and the other five factorial will be 5 times 4 times 3 times 2 times 1 and we can say 10/5 is 2. 9/3 is 3. 8/4 is 2. 6/2 is 3 and 7. So we have 2x3x2x7x3. Which is equal 252. [BLANK AUDIO]