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Suppose you walk into a casino. You will see all sorts of games varying from blackjack and poker to slot machines. It would not take long for you to notice that there are some players who are winning some money, sometimes a substantial amount. You might wonder how the casino makes money when they are clearly giving some money away.
Casinos have a clear understanding of expected value. The expected value for a situation can be thought of as the average result over the long run. In other words, it can be thought of as the expected winnings or average payout for a game of chance. Consider the thinking of the owner of a casino. While there are some people who win a little, and occasionally a few people who win a lot, most people end up losing some money at the casino. The casino actually expects some people to occasionally win big. In fact it makes for great advertisement! As long as the mathematics show that the expected value is in the casino's favor, the casino will continue to make money in the long run. In this section, we will focus on how to calculate the expected value.
As mentioned above, the expected value can be thought of as the average result over the long run. Recall that to find the average value of a series of numbers, we simply add up the numbers and divide by how ever many numbers there are. For example, the average of 3, 4, 5, and 6 is 4.5 because
You will notice that the average value of 4.5 is not one of the numbers in the original set of numbers. This is often also true with expected values. The expected value for a situation does not have to be one of the possible values.
Use the concept of averages to find the expected value for the example below.
Example 1
A game is played in which a coin is flipped one time. If the coin lands on tails, the player wins $5. If the coin lands on heads, the player wins $10. What is the expected value for a player who plays this game one time?
Solution
The expected value is $7.50. This is strange because it is actually impossible for a player to win $7.50. They could only win either $5 or $10 but the average win will be $7.50. One way to see this is to actually play the game two times. If the flips come out matching their theoretical probabilities, one of the flips will be heads for $5 and the other will be tails for $10. The player will have won $15 in two games so the average win or expected value would be
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