Some People are Better Off Playing the Lottery than Others – and You’ll Never Guess Why
If I asked you how much you would pay to play the lottery, what would your answer be? To answer this question correctly, we must first understand the concept of ‘expected value’. Expected value can be thought of as the average level of a random variable based on its probability distribution over the long-run.
Let’s give an example. Assume I offer you the chance to play a game. In the game, I will flip a fair coin. Every time the coin is heads you win 10 $ from me, and every time the coin is tails, I win 5 $ from you. From your perspective, the expected value of this game is +2.5 $, since that is what you would be expected to win in the long run; if you were rational, you would be looking to play the game for less than that amount.
Formally, this looks like: (.5)10 + (.5)-5 = +2.5
The lottery is like this game, just with bigger numbers and other considerations like taxes and investment potential.
Basically, this formula would look something like this:
EV = (p(Jackpot) * subsequent prize + investment potential when won - taxes) – (p(loss) * cost of ticket)
To properly dole out the numbers of this calculation would take some rather involved math, and we will rightly avoid that here. Understanding the formula conceptually, however, offers many interesting insights by its own accord. Firstly, if we bring our attention to the investment potential variable, we might ask what would affect the size of this variable.
Looking at a simple compounding interest formula is useful here:
A = P(1 + r)^t
Where A=amount, P=principal amount, r=annual interest rate, t=time in years
Relating this to the lottery, P = Jackpot-taxes, r=average investment return rate, t=life expectancy in years
Thus, a fascinating revelation emerges: the younger you are, the higher your expected value is for the lottery.
It is important to note that, while there are rare instances where the lottery can be +EV, that does not mean you should play it. For example, if you have 1000$ to invest, and you invest 100$ into the lottery every time it is +EV, you are expected to go broke. EV calculations do not take into consideration the finiteness of investment portfolios. In fact, you would have to play the Powerball lottery 292,201,338 times to expect (no guarantee) to win it. If you bought a ticket once a week, you would need about 73,000 lifetimes to expect to win it. Not to mention, there are various other investments available to you that are higher EV than playing the lottery.
If you do decide to play, you might find this simple site I made useful in tracking when you should play. Make sure you do your own due diligence though; the site only tracks the pre-tax annuit jackpot. And please, gamble responsibly.