1. MAT 1000 Mathematics in Today's World

2. Last Time

5. Today We will introduce another counting formula. We will discuss the idea of “expected value,” also called “probability mean.” Expected value is the reason casinos and insurance companies can turn a profit. They rely on the “law of large numbers.”

6. Counting

7. We are trying to count ways of filling five “spaces” ____ ____ ____ ____ ____ In each space, we have 39 choices, and we don’t repeat. But we can’t use the formula for permutations. In a four-digit PIN or a list of gold, silver, bronze medal winners, the order of the list matters. The PIN 1234 is not the same as the PIN 2413. This isn’t true of lottery numbers.

8. Counting

20. It’s a little hard to understand probabilities this small. So here’s a way to make this more concrete. 575,757 is roughly how many seconds there are in one week. Imagine that I choose one specific second over the next seven days. Winning the Fantasy 5 lottery is like picking that exact second.

21. Expected value Of course if 500,000 people play, then we do expect someone to win. The lottery is daily, and the jackpot is at least $100,000 (it varies). But even though someone probably wins every day, you don’t need to worry about the state of Michigan. Just like casinos and insurance companies, they know how to price the tickets so that they will always make money. How do they do that? Using expected value, which your textbook calls the probability mean.

22. Expected value

25. The law of large numbers

26. This is fairly useless information to you. If you had enough money to play the lottery hundreds of thousands of times, why would you? But this is very important to the state of Michigan. Over the course of a year, millions of people will play the Fantasy 5. According to the law of large numbers, Michigan can expect to pay out on average of $0.17 per player. So as long as a lotto ticket costs more than $0.17, they will make a profit.

27. The law of large numbers The law of large numbers is also important for casinos. Again, not for you as a player. You can’t play enough times for it to matter. But there will be millions of gambles made at the casino, and with that many gambles, the average payout will be very close to the expected value. So they can rely on the law of large numbers when they set prices, and they know they will make a profit.

28. Expected value Here’s another example of using expected values. At a carnival, you pay $2.00 to play a coin-flipping game with three fair coins. On each coin, one side has the number 0, and the other side has the number 1. You flip three coins, and you win $1.00 for each 1 that appears on top. How do your expected earnings compare to the cost of the game?

29. Expected value First, let’s look at the probability model for tossing these three coins. We can think of outcomes as lists of three numbers, which are either 1 or 0. In other words, we are choosing three things from two candidates. Notice we can repeat (all of the coins can come up 1 or 0).

30. Expected value

31. 000100010001 110101011111

32. Expected value This is the probability model for tossing three coins. Based on this we can write down a probability model for the game. The outcome of the game is amount of money we win, which equals the number of coins which show a 1. So the sample space is {0,1, 2, 3}

33. Expected value

34. OutcomeProbability Win $01/8 Win $13/8 Win $23/8 Win $31/8

35. Expected value