Probability & Statistics – Page 11

Expected value (also called mean) is often used in probability and statistics to help describe data sets or to predict outcomes. However, this raises the question of what types of values we can get for expected value. So, can expected value be negative? Expected value can be negative. However, a negative expected value is only possible if some of the data or outcomes have negative values. This is because probabilities are never negative. Of course, expected value can also be positive or zero. No matter the sign, expected value is a tool, and it does not always indicate a possible or likely outcome. In this article, we’ll talk about negative expected value and what it means. We’ll also give some examples of negative expected value to help shed some light on the idea. Let’s begin. Can Expected Value Be Negative? Expected value can be negative in certain cases. However, this requires at least some of the data or outcomes to have negative values. The reason is that probabilities are never negative, and expected value comes from adding up products of probabilities and outcomes. (Also, probability is never greater than 1 - you can learn more in my article here). Remember that expected value is also called mean – they are the same thing. However, the most common type of mean that we are familiar with (arithmetic mean) is not always the correct way to calculate expected value. In fact, arithmetic mean is a weighted average that assumes equal weight for all data points. In other words, arithmetic mean assumes the same probability for each possible outcome (we add up all the values and divide by how many values there are - an implicit weight or probability of 1/n for each of the n data points). Let’s look at some examples to illustrate how negative expected values can occur (and when they cannot occur). Example 1: Negative Expected Value RazeFunding, Inc. is a tech company that recently had its IPO. You can buy a share of this company for $100. Expected value is a tool that can help you to decide whether to invest or to take other actions that involve risk and reward. You want to find the expected value of your return on this stock in one year. You believe that there are two possibilities for the value of a share in one year: 1.) The value rises to $300, an increase of $200 (this is unlikely, with only a 10% probability of such a large gain).2.) The value falls to $10, a decrease of $90 (this is very likely, with a staggering 90% probability of such a drop). Remember that to find the expected value of your return on the share, you multiply the dollar value (gain or loss) for an outcome and the probability of that outcome. Then, we add up all of those results. The calculations and table below will help to make it clearer. Outcome 1: the value of the share increases by $200. There is a 10% chance (0.10 as a decimal) of this happening, so we calculate: (+$200)*(0.10) = +$20. Outcome 2: the value of the share decreases by $90. There is a 90% chance (0.90 as a decimal) of this happening, so we calculate: (-$90)*(0.9) = -$81. Adding these two results together, we get $20 - $81 = -$61. So, the expected value of your return in a year if you buy the share is -$61. EventValueProbabilityProductShareup to$300+20010%(0.10)+20Sharedownto $10-9090%(0.90)-81TotalNA100%(1.00)-61(exp. value)This table summarizes value, probability, andtheir product for each event, along with theexpected value at the bottom right. Note that the expected value of -$61 is not an outcome that can actually occur. You either gain $200 (if the share value goes up to $300) or you lose $90 (if the share value goes down to $10). Instead, the expected value is a decision-making tool. It can help us to decide whether to pursue an opportunity or not. Applying the concept of expected value over the long term can help us to achieve a positive overall return, even if we have some losses and failures along the way. However, you will lose money in the long run if you only pursue opportunities with negative expected value. You may win sometimes, but overall, you will lose more than you gain. Example 2: Negative Expected Value Let’s look at an example that is a little more involved. Assume that we have a fair die with 6 sides, labeled 1, 2, 3, 4, 5, and 6. There is a 1/6 probability of rolling each number from 1 to 6. Rolling one or more dice has an element of chance, and expected value can help you to analyze the possible outcomes. The game will be as follows: if you roll a 6, I will pay you $18. Otherwise, you will pay me $6. Let’s find the expected value of this game for you. The outcomes and probabilities are as follows: Outcome 1: Win $18 (+$18) – this only happens if you roll a 6 (a 1/6 probability).Outcome 2: Lose $6 (-$6) – this happens if you roll 1, 2, 3, 4, or 5 (a 5/6 probability). Taking the product of each outcome times its probability, we get: Outcome 1: (+$18)(1/6) = +$3Outcome 2: (-$6)(5/6) = -$5 Adding up these two results, we get: +$3 - $5 = -$2. So, the expected value of playing this game is -$2 for you. If you were to play 18 games, you would expect to lose $36: (18)(-$2) = -$36. On the flip side, the expected value of playing this game is $2 for me. If I were to play 18 games, I would expect to gain $36: (18)($2) = $36. In other words, your loss is my gain in this game. EventValueProbabilityProductRoll 6+181/6+3Roll 1,2, 3, 4,5, or 6-65/6-5TotalNA6/6-2This table summarizes value, probability, andtheir product for each event, along with theexpected value at the bottom right. So, what would your winnings on rolling a 6 have to be to make this game “fair”, so the expected value of playing it is $0 for both of us? Let’s call the value X. Then we want to solve for expected value = 0, or: (X)(1/6) + (-6)(5/6) = 0 X/6 – 6 = 0 X/6 = 6 X = 36 So, you would have to win $36 when you roll a 6 in order to make this game fair (an expected value of zero). If we played this game many times in a row, we should both expect to break even (or close to it). Can The Mean Of A Normal Distribution Be Negative? The mean of a normal distribution can be negative. This can happen if some or all of the data points have negative values. For example, let’s say that the weight change of a person over the course of a year has a normal distribution. If the average person loses weight, then the mean (expected value) might be -2 pounds, and the standard deviation could be 4 pounds. Although some people will gain weight, more people will lose weight. You can find out more about when a normal distribution is negative (and which parameter cannot be!) in my article here. What Does A Negative Expected Value Mean? Negative expected value means that you can expect a loss (negative value) if you continue the same trial (or pursue the same opportunity) over and over again. Thinking back to the tech company from before: let’s say that there are 1000 such companies, all with the same current value, potential outcomes, and risk profile. If you invest $100 in each of those 1000 companies, then you expect to lose $61 from each investment (since the expected value is -$61). That means that your $100,000 investment will be reduced to $39,000 in a year (a -61% return!) How does this happen? Let’s look at the numbers. You invest $100 in each of 1000 companies, for a total investment of $100,000. For 1000 companies, we would expect 10% of them (100 out of 1000 companies) to succeed. That means the share price would increase to $300 for those 100 those companies. The value of those shares, in total, would be ($300)*(100 companies) = $30,000. For the other 90% (900 out of 1000 companies), the share price would decrease to $10. The value of those shares, in total, would be ($10)*(900 companies) = $9,000. Adding these values together, your investment portfolio is worth $30,000 + $9,000 = $39,000. I don’t know about you, but I don’t want to lose 61% of any portfolio – and certainly not in a single year! Conclusion Now you know a little more about negative expected value and how it can happen. You also know exactly what a negative expected value means and how this tool can help us to make better decisions. You can learn more about mean (expected value) and median in my article here. You can learn more about the uses of expected value in my article here. I hope you found this article helpful. If so, please share it with someone who can use the information. Don't forget to subscribe to my YouTube channel & get updates on new math videos! ~Jonathon