 Today, we're going to discuss probability, and we're going to look at the topic of expected value. To discuss expected value, let's look at a friendly game of chance. Here's the question. Your friend challenges you to a friendly game of chance. It costs $1 to play. If you win, you get $2, and all you have to do to win is roll a six-sided dice and roll a five or a six. How much money should you expect to win? I have win here in quotation marks, and we'll see why in a second. The expected value is how much you would expect to win on average over time. Let's say you play this game 1,000 times. You play this game 10,000 times. On average, how much are you winning? How much are you losing each game? To look at expected value, we take a look at the probability of winning times the profit. Not how much you win, but how much you profit from your winnings, and then subtract from that the probability of losing how much you lose your loss. When we take a look at this problem here, you need to roll a five or a six to win. There are two ways to succeed out of six possible outcomes. You have a one and three chance of winning. I'm going to go ahead and plug that in. The probability of winning would be one out of three. If you don't win, you lose. Using the complement rule, if you have a one and three chance of winning, you have a two and three chance of not winning of losing. Now all we need to figure out is how much you would profit from winning and how much you would lose from losing. You win two dollars, but it costs you a dollar to play the game. You need to do two minus one to determine that you would profit a single dollar by playing. We would do one third times a dollar, we would profit a dollar minus, I'm going to move my subtraction sign a little bit closer here, minus two thirds times, and it costs a dollar to play the game. Times one. One third times one minus two thirds times one. If we were to plug and chug this, you would end up with a negative one third. Since we're dealing with money here, I'm going to say a negative 0.33. On average, you would lose 33 cents. That's what the negative 0.33 would say. If you play the game 100 times, 1,000 times, 10,000 times, in a single game, you would never lose 33 cents, but on average, over time, you would lose about 33 cents. Playing this friendly game would also mean that your friend is profiting on average 33 cents. They're probably a happy person, and after playing for a few hours, you might not be a happy person yourself, so that's something to consider.