Expected Value Explained - Should You Play This Game?

In this video, we dive into the concept of expected value to help you navigate choices involving probabilities and payouts. Imagine your friend offers to play a game with a six-sided die: if you roll a 1 or 2, they’ll give you $48, but if you roll anything else, you’ll have to give them $30. Should you take the risk? We break down how to calculate the expected value to determine if this game is worth playing—and how this concept applies to other real-world decisions like investments. Stay tuned as we walk through examples of calculating expected value for different scenarios, so you can make more informed choices in the future. Whether you're considering small bets or major financial decisions, understanding expected value can be a game changer! Transcript: Your friend offers to play a game with you involving a six-sided die. If the roll is 1 or 2, they will give you $48. If it’s anything else, you have to give them $30. Should you play the game? To decide whether or not you should play, and how to navigate various choices involving probabilities and payouts, we can use the concept of expected value. Expected value is the average outcome of a single instance of an event over the long run. For example, let’s say we flip a coin. If it’s heads, you get $10; if it’s tails, you get nothing. The expected value is $5 because your winnings would average out to $5 per flip. To calculate the expected value for an event, decision, or game, start by identifying each of the possible outcomes and the probability of each. We are essentially calculating a weighted average. For our coin flip example, there are two outcomes: $10 (heads) and $0 (tails). The probability of each is ½. Then, we multiply each outcome by its probability of occurring and add the results. $10 times ½ is $5, and $0 times ½ is $0. Add them together, and we get $5 as the expected value for a single flip. We can use expected value to estimate the outcomes from playing many times. For example, if we played the coin flip game 20 times, we’d expect to end up with 20 x $5, or $100. This makes sense, right? If we play 20 times, we expect to get heads 10 times, earning $10 for each of those heads. Now, let’s look at our original example: the game where we roll a die. If we get 1 or 2, we get $48, and if we roll a 3, 4, 5, or 6, we lose $30. The two outcomes are $48 and -$30. Because all rolls of the die are equally likely, we can add up the payouts for each scenario and divide by 6. In other words: The probability of getting $48 is 2 out of 6, and the probability of losing $30 is 4 out of 6. $48 x 2/6 = $16, and -$30 x 4/6 is -$20. We add them together and get -$4. So, our expected value is -$4. On average, we will lose $4 for each play of this game. The general formula for expected value is the sum of each outcome multiplied by the probability of that outcome occurring. One additional note: sometimes an expected value in our favor isn’t enough for us to participate. This occurs if any of the outcomes are unacceptable to us and we don’t have the option to play multiple times. Imagine a $6,000 investment with an estimated 10% chance of getting back your $6,000 plus a gain of $200,000, a 10% chance of getting back your initial investment of $6,000 and coming out even, but an 80% chance of losing the entire $6,000. The expected value is $15,200, which is substantial, but you may not want to open yourself to the most likely outcome of losing $6,000. Thanks for watching! See you next time.