It is hard to avoid hearing about the $1.5 Billion Powerball. Along with the size of the prize, there has been a lot of attention on your chances of winning (1 in 292 million). So, I thought this was as good as any excuse to talk about the expected value. Here are three lessons:
#1: “Wait, Is the Lottery Now a Good Investment?” (Or, How do you calculate the expected value). If you just read the headlines and calculate the expected value, you come up with a nice return. To calculate the expected value of your winnings you just need to multiple the chance of winning (1 divided by 292 million) by the payout ($1.5 billion). This simple calculation yields an expected win of $5.14. So, based on this, every $2 ticket you would buy would yield an expected return of $5.14. This seems like a great investment. (In fact, this is why you see articles on why it might be a good idea to buy all possible combinations.). But, this may be too good to be true….
#2: “Whoa, Did our Formula Capture all the Details?” (Or, don’t forget to get all the costs and benefits included in the analysis). There are two easy things to fix about the $1.5 billion. First, the advertised $1.5 billion is not expressed in Net Present Value (NPV) terms. According to Powerball’s homepage, the cash value is $930 million. The second easy thing is that we need to take about 50% away from the $930 million for taxes. This lowers us to about $465 million an an expected return of about $1.60. But, the last thing we need to consider, and the lesson for businesses, is that we need to consider what others will do. In the case of the lottery, there is a very good chance that there will be more than one winner. Once you start splitting the jackpot, the expected value decreases quickly. (For a more detailed analysis, see FiveThirtyEight.)
For a business lesson, this last point reminds us to include information on how our competitors will react when we build a business case. For example, if we come up with a new pricing strategy that predicts new sales, we should also factor in that our competitors may also adjust their strategy.
And, to wrap up, the last, and I think most relevant reminder for the business community…
#3: Wow, Expected Value Sure Does Hide A Lot. (Or, you should pay attention to variability when making decisions.) In this case, the expected return on your $2 lottery ticket is about $1.32. But, even it had come up as $2.50, it feels like we are completely missing an important part of the story. The winner of the lottery is going to see hundreds of millions of dollars and the losers are going to be out of each $2 bets they made. In other words, it is not like everyone who plays will each take home $1.32 from their bets. In many cases, like this one, variability is the big missing variable.
Toyota’s famed lean manufacturing system relied on removing variability to create a new way to think about manufacturing. Sam Savage wrote a book called the Flaw of Averages that talks about the danger in using just the average and not some measure of variability when making business decisions. And, we’ve written articles on how to buffer variability in your operations (see here and here).