After watching this video lesson, you will understand what expected value means. For example, if the probability of rolling a 5 in a dice game and winning $10 is 1/6, we would multiply the $10 (the event) with 1/6 (the probability of it happening), and then add it to all our other choices. In this case, you follow the same format as we did for the dice game (multiplying the event with its probability and summing them all up), but instead of all the probabilities being 1/6, they will each be different. In real life, you’re likely to encounter more complex expected values that have more than two possibilities. You can increase the cost per game or decrease the possible earnings. You toss a fair coin three times. Why won’t people risk a lot of money if the odds are certainly in their favor? The probability (P) of getting a question right if you guess: .25 The formula changes slightly according to what kinds of events are happening. You can use this value to determine whether a game is worth playing or not. Step 1: Insert your “x” values into the first few values for the formula, one by one. (P(x) * n). Your first 30 minutes with a Chegg tutor is free! The probability density function is f(x) = 1/2x. For a step-by-step guide to calculating this, see: Expected Value. But despite that fact, people aren’t willing to pay much money to play it. So, what do you do? Log in here for access. In other words, the function must stop at a particular value. Enrolling in a course lets you earn progress by passing quizzes and exams. There has to be something wrong with the game’s odds. Of course, calculating expected value (EV) gets more complicated in real life. He has performed as Teacher's Assistant and Assistant Lecturer in University. Two thousand tickets are sold. They calculate what is called the expected value of the game. Visit the Business 110: Business Math page to learn more. You’ll note now that because you have 3 prizes, you have 3 chances of winning, so your chance of losing decreases to 197/200. For example, in a game of roulette, they need to know the probability or chance of the ball landing in each of the numbers on the roulette wheel. How much would you bet if you could always win? Suppose for $1 you choose six numbers from 1 to 48. All rights reserved. The Paradox is this: There’s a simple betting game you can play where your winnings are always going to be bigger than the amount of money you bet. Because this expected value is an average, you can expect to hit this number when playing the game. That’s a losing proposition for you (although the school will rake it in). This section explains how to figure out the expected value for a single item (like purchasing a single raffle ticket) and what to do if you have multiple items. You can think of an expected value as a mean, or average, for a probability distribution. If the expected value of a game is -$5.00, for example, you might say it's not worth losing $5 to play it, or you might say that the fun of playing the game is worth losing $5. Would you play? To calculate this value, you multiply each event with its probability and add them all up. If the expected value is negative, then the player loses money and the casino gains money. Perform the steps exactly as above. The probabilities are: 1/8 for 0 heads, 3/8 for 1 head, 3/8 for two heads, and 1/8 for 3 heads. They aren’t willing to risk their money even for a sure bet. As a member, you'll also get unlimited access to over 83,000 For example, if the expected value of playing a game is -$1, you can expect to lose a dollar each game as you keep playing more and more games, even if your possible wins are only $0 and $10. Chegg.com will match you with an online tutor, and your first 30 minutes is free! Of course, if you hit the jackpot and win a million dollars from just one game, you would think that these games are great! imaginable degree, area of A discrete random variable is a random variable that can only take on a certain number of values . With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Therefore, the general formul… Expected Value and the Lottery . For example, if you were rolling a die, it can only have the set of numbers {1,2,3,4,5,6}. How do they do this? Since it measures the mean, it should come as no surprise that this formula is derived from that of the mean. Step 1: Find the mean. Get access risk-free for 30 days, The expected value is the average value you can expect after a large number of rounds. Let’s say your school is raffling off a season pass to a local theme park, and that value is $200. Step 2: Figure out your probability of getting each value of X. Example of Expected Value (EV) To calculate the EV for a single discrete random variable, you must multiply the value of the variable by the probability of that value occurring.