The Calculated Gamble: When Does It Make Sense to Buy a Lottery Ticket?
As we are in the midst of lottery fever again and I would like to capitalize on the free hype I have chosen to finally write this post.
I occasionally play the Powerball. But since I like to think I am smarter than the average Powerball patron I tell myself I only play the powerball rationally. People assume that playing the lottery is always an absurd thing to do. Most jeer that the only people who play the lottery are those that are mathematically illiterate. But I only play when the math is on my side.
Let’s cover the basics. There are 292,201,338 potential winning combinations. 5 white balls drawn from a pool of 69 numbers and then one red ball drawing from a pool of 26 numbers. We plug this into the following combination formula:
As we have two distinct pools: red balls and white balls, we must use the formula twice.
Now the first is a little more complicated as you are drawing multiple balls from on pool but the second is obvious. If you have 1 ball to draw from a pool of 26 chances you have a 1/26 chance.
To win the overall prize you must match both. In the same way hitting heads twice when flipping a coin requires you to multiple 1/2 by 1/2 to get a probability of 1/4 so here you must multiple 1/11,238,513 by 1/26 to get the 1/292,201,338 chance at winning the powerball.
So now I ask you at what prize level should a rational person buy a ticket? Is there such a price or is the lottery always for suckers. At the price of $2 a ticket for you to be getting fair odds you need a jackpot size of $2*292,201,338 = 584,402,676. If you haven’t checked the latest jackpot it is $900 Million. So perhaps you should consider buying one.
I have left out two important considerations: taxes and the annuity vs. cash option. Let’s assume that our tax rate upon winning is 37%, the current top marginal tax rate in the United States. It is true that this only applies to income of over $578,126 if single and $693,751 if married but as this will constitute far less than .1% of your 900 million in winnings it is a rounding error. It pays to get married before you win the powerball when it comes to taxes but should divorce ensue it will be a costly endeavor indeed.
Taking the 37% marginal rate into account you actually need the lottery to be ($2*292,201,338)/(1-.37) = $927,623,295. So at 900 million we are almost at the right time to buy.
But wait we still have yet to discuss the cash vs. annuity option. The number you see plastered on billboards during your grueling commute home from your soul sucking corporate job is the annuity amount.
For the annuity option the lottery corporation takes the cash prize and buys risk free government bonds for you and pays you a certain amount once per year over the next 29 years. 29 years as you receive the first portion immediately after winning. Imagine if you had to wait a year to receive it.
How can they guarantee the payments over such a long period of time? They are legally obligated to invest solely in US government debt. If you buy some of uncle sam’s debt he will reward you with a monopoly on gambling in your jurisdiction. Currently the 30 yr. bond is trading at 3.9% rate. This means that if you buy a 30 year bond every year for the next 30 years you shall be paid 3.9% per year.
So what rate of return is baked into the annuity option? Now we must do a discount cash flow analysis, the formula for which I will spare you. What annual rate of return is required to turn 465 Million today in 900 million 29 years from now. A measly 2.3%.
So the lottery company is taking a 1.6% (3.9–2.3) spread off the top every year for the next 29 years amounting to around 500 million total. Or over 16 million per year.
If the average person buying your tickets is an idiot maybe the average person who wins will be too. Powerball is giving themselves $500 million for buying some bonds and holding the money for you. Not bad for a minute’s work.
So now that we have proven the annuity option is always for suckers what does the cash prize need to be for it to be worth buying a ticket. We already calculated ($2*292,201,338)/(1-.37) = $927,623,295, and as the annuity is always almost exactly double the cash prize we can multiple by two.
$927,623,295*2 = $1.86 Billion. As there has only ever been one jackpot over this amount, 2.04 billion in Nov 2022, it is rarely optimal to buy a ticket. Was this the only drawing where is was rational to buy a ticket?
We are still not done. More than one person can win the prize if they both pick the winning numbers. So how often do multiple people win? If you look at all drawings rarely does more than one person win. But as we are only looking at very large jackpots we need to shrink our sample. This is due to more people playing the larger the number gets.
There have been 17 wins of over 500 million in powerball history. The average number of winners for these was 1.53 people: 26 wins for 17 jackpots. For three of these there were three individual winners. I feel sorry for these poor souls having to split their prize worth hundreds of millions with others.
$1.86 Billion multiplied by 1.53 winners on average = $2.85 Billion. In Powerball’s 28 year history consisting of 4,368 drawings there has never once been a rational time to buy a ticket. I lied to you. Thanks for playing!
Last year powerball sold around $50 million dollars per week for a total of 7.8 Billion for the year. In the two weeks leading up to the 2.04 billion Jackpot they sold over $2 billion worth alone.
This exercise in logic is futile. People do not buy lottery tickets for rational reasons. People spend more than $80 billion on lottery tickets of various kinds per year. That’s more money than people spend on books, sports tickets, video games, music, and movie tickets combined.
They do this because they like to dream. Spending $2 for the chance of imaging what you would do with that outrageous number on the billboard shown on your afternoon commute after a long day is more than enough compensation.
I would go so far as to say it is the most rewarding form of gambling, assuming you only buy 1. You are not wasting hours at the blackjack or craps table where the more you play the more likely you are to lose. Purchasing a ticket takes a mere second. Well, assuming there isn’t a huge line due to a massive jackpot. I feel sorry for gas station clerks everywhere. I’ll tell you this if my numbers get hit tomorrow you may not hear from me again.
