 Hi, I'm Josh Grab and today I'm going to talk about a probability concept called expected value. What do you expect will happen if you do something over and over and over again? What do you expect will be the average result? That's kind of what we want to think about as we talk about expected value. Let's use an example that is pretty common to everyday life playing the lottery. Suppose you buy a lottery ticket for $5. You get one play of the lottery for $5 and you're playing a particular lottery game that has a payoff if you win of $1,000. That's if you win, what if you lose? If you don't lose $1,000, you just lose $5. The $5 you used to buy the ticket, they just take that and keep it. Don't pay off, that's money in your pocket, we'll leave that positive but the $5 loss we should indicate with a negative, you're losing that. If you had an equal chance of winning and losing, you could play once today and lose and play once tomorrow and win and you would win $1,000 today and lose $5 tomorrow and you would have a total of $995 and on average you've played twice so you would have almost $500 per play of the lottery if you did that. It's not very likely that you'll play twice and win once. The lottery, it's not going to stay in business very long if that's what happens. Every time you play twice, you won once. If everyone did that, the lottery can't stay in business. The lottery has to change the payout, make it less likely that you'll win. This is the particular lottery game that you're playing has a 1 in 1,000 probability of winning. Again, this is if you win front, you win $1,000. If you win, the probability of winning is 1 in 1,000. If you lose $5, negative $5, you win negative $5 and the probability of losing is all the other times out of 1,000 that you didn't win, $199 times out of 1,000. Well, let's suppose you played the lottery 1,000 days in a row. You pay your $5 and one time you win, one time and you win $1,000. Put your winnings over here. You won one time, you win $1,000. That's 1,000 times one. And you lost $5,999 times. We want to figure out what your total winnings or your total losings in this case turn out to be. And then we'll talk about your average winnings or average losings. We do this subtraction and multiply that out and we get 1,000 times 1, 5 times 999 is 4,995. But I've lost that. In total, I'm in the hole. I've lost $3,995. But that's not per try at the lottery. That's $3,995 over 1,000 plays, 1,000 tickets you purchased. What does that work out to be? On average, we've lost $3,995. But we played 1,000 times. We do that division. On average, we've lost $3,995. Or we might just round that up to $4. On average, we call that expected value. If you play over and over and over again, on average, you're losing $4 every time you play. Each individual time you're losing $5 or winning $1,000. That's not what expected value looks at. It doesn't look at individual times you played. Expected value is, more properly, in long run average. If you do it over and over and over again, thousands of times, you would expect that, on average, you're losing about $4 each time you play. The lottery company realizes this. Every time you lose $4, they make $4. And they can continue to support their business and sell you lottery tickets. And occasionally, you might win. Some people will win more often than they won't lose as much. They might even have positive winnings over their lifetime. But there will be more people that lose. If somebody wins more than this, on average, for over their lifetime, there will be other people to balance it out. On average, people are losing $4 every time they play this $5 lottery. And that's expected value.