 Today, we're talking about probability and we're looking at the topic of expected value and we're looking at raffle drawings. So if at this point in your life you have not experienced at some point, I'm sure you will, someone, perhaps an adorable small child, asking you to buy a raffle ticket for some kind of fundraiser. In the small town I live, this happens regularly. So let's look at the question. In a small town, the local chapter of American Legion is holding a fundraiser raffle to pay for the upcoming county fair. They have printed a thousand raffle tickets to be priced at $10 each. There will be several drawings for winning. If you win, your ticket is removed from the drawing so you can only win once. If you win first place, that would be the first drawing, you win $1,000. If you win second place, you win $500. If your ticket is drawn third, you win $250. If your ticket is drawn fourth, you win $100. And if your ticket is drawn fifth, fifth is the last winning ticket, you would win $50. So the local community in this town is a fine, the local community in the small town finds the winnings very favorable and quickly buys up all 1,000 tickets. So looking at the jackpot, some wonder if the American Legion will even turn a profit on this fundraiser. So your mission is to determine the expected value for a raffle ticket and also to determine how much the American Legion is profiting from this fundraiser. Alright. So there are 1,000 raffle tickets priced at $10 each. So what is the probability of winning the $1,000? Only one person will win the $1,000 out of the 1,000 raffle tickets sold. And how much, so this will be the probability in this here, let's talk about profit. How much would you profit by winning the $1,000? You wouldn't profit the full $1,000 because you had to pay $10 for the ticket. You would end up profiting only $990. Similarly with the second place prize, you would win $500, but it cost $10 to play. You would profit $490. Now once the first place ticket has been drawn, it's removed. So when the second place drawing takes place, there are only 999 tickets left of which one will win the second place prize. So we could go through and look at the probability for each one of these as well as the profit, but the screen is only so big here. So I decided to go ahead and put it in a nice table here. So you have an outcome of winning first place, the probability of that outcome occurring, and then you have the profit, so on and so forth. And then the bottom row right here is the outcome of losing. So five people get to win, 995 people have to lose. So the probability of losing the raffle, not winning anything, would be 995 over 1,000 and those people lose $10, the price of the ticket. So knowing what we know about expected value, to determine the expected value, we would take a look at the probability of winning first times profit, plus the probability of winning second times profit, and then the probability of third, fourth, and fifth, since I don't have the room on the board here, I'm not going to write that out. And then at the end, minus the probability of losing times the loss. And as you see, we're running out of room on the screen here, so I figured I would take the time to write this down. So you see here the probability of finishing first times the profit from the first place prize, plus the probability of finishing second times the profit for the second place prize, plus the probability of finishing third times third place profit, plus the probability of fourth times fourth place profit, plus the probability of fifth times fifth place profit, plus, and what I did here, I did, plus the probability of losing times your loss, and I wrote it as a negative. So when you plug and chug this, you end up with an expected value of negative eight dollars and 10 cents. So this raffle drawing that the local community found very favorable, that the local community bought up very quickly, perhaps they posted their lottery ticket or their raffle ticket on the refrigerator, an expectation of winning money. The expected value for that ticket is eight dollars, negative eight dollars and 10 cents. So one way you could articulate this is if you play this raffle a hundred times, a thousand times, 10,000 times, on average you would lose eight dollars and 10 cents every time you play. So it's not a very profitable raffle for the player, but is it profitable for American Legion the organization putting the fundraiser on? So American Legion would have to pay out $1,900 in winnings. I came up with this number by taking how much you would pay out the winnings for first place, second place, third place, fourth place, fifth place. They would pay out total a little less than $2,000. They would earn $10,000 in revenue, remember $10 per ticket, sold a thousand tickets, so that's $10,000 total dollars. So off of this fundraiser, American Legion would raise just over $8,000 to go towards next year's county fair. And as someone that's worked with nonprofits in the past, anytime you get a fundraiser anywhere near that, typically pretty thrilled. So this raffle is great for American Legion, but maybe not so great for you if this is your retirement plan.