Let’s take a look at the odds of winning the lottery. To keep things simple, we’re going to be looking at the Mega Millions lottery (the biggest multi-state game in the U.S.). Here are the statistics from the Mega Millions website, describing where the money from lottery proceeds goes (the exact numbers vary by state):
Where The Money Goes (Avg.)
35%- State Government Services (primarily education)
50%- Prize Money
15%- Retailers & Operating Costs
Thus, on the average (over time) lottery ticket purchased, a person will get back 50% of the cost of the ticket (or put another way, they will lose half of their money).
Granted, this “average” for lottery odds is somewhat deceptive. The average is skewed by the top-heavy distribution of the Jackpot (a very small number of people getting a very large payout). Nonetheless, that number (the average payout for a ticket purchased) is the one I am focusing on.
So, since we have established the average payout of 50%, this begs the question: is it ever a good investment to purchase a lottery ticket?
The answer most people will give is NO. I think it is safe to say that the typical lotto ticket purchaser doesn’t really expect to get his or her money back. What people are really concerned about is that extremely miniscule chance of hitting the big number- the Jackpot. Thus, to them, it’s not about average returns.
It’s about the reality that there is a set of 6 numbers on any given Tuesday or Friday that, if written down on their lottery receipt, would make them unimaginably rich. And in most cases, this belief about average payouts is true.
However, the reality is that (as with all averages) there are occasions where actual payouts are significantly different than the averages. Thus, we must make a differentiation when using the word “average”. Here are the two ways we are using it:
Over Time Average Payout: all of the money paid out divided by all of the money spent by players over a long period of time and across many drawings (these are the #s listed above)
Single Drawing Average Payout: all of the money paid out divided by all of the money spent by players given one drawing’s particular set of variables (which we will look at later)
Thus, while it is true that the Over Time Average Payout stays very close to 50%, the Single Drawing Average Payout can fluctuate a substantial amount. The point I am trying to make is that we, as players, should not be nearly as concerned with the Over Time Average Payout lottery odds as we should be with the Single Drawing Average Payout lottery odds. The reason for this is that most people don’t purchase a ticket for every possible drawing- but rather a small number of drawings, or perhaps even one.
There is one primary variable that determines Single Drawing Average Payout (and thus, why it differs from the Over Time Average Payout), and that is the Estimated Jackpot. Our formula says that Average Payout = Total Money Paid Out / Total Money Spent. Thus, we can see that a winning Jackpot of $30 Million vs. a winning Jackpot of $300 Million give us significantly different Average Payouts (Single Drawing).
For Mega Millions, there are 175,711,536 possible combinations of numbers (5 #s selected from 56 possible + 1 # selected from 46 possible). So, the Jackpot Odds (the odds that you will get all 6 numbers correct) are 1 in 175,711,536.
Given that, one could argue (and many do) that once the Estimated Jackpot surpasses $175,711,536, the Single Drawing Average Payout is greater than $1.00 (the cost of a ticket). In fact, considering these factors alone, the SDAP exceeds $1.00 prior to the EJ surpassing $175 Million because you also have to consider all of the smaller prizes.
Now for the bad news.
While it is true, as we have demonstrated, that you can significantly improve your odds even to the point where your Single Drawing Average Payout is greater than the cost of the ticket, this is not the whole story. In reality, 2 other factors ultimately negate much of the improved odds we have discussed. First, the Jackpot amount listed is what a winner would receive only if he or she chooses to receive the amount in annual paymentsover a period of 25 years (in most states). The present value of that Jackpot is generally a little more than half of the listed payout. Furthermore, federal and state taxes cut out another 50% or so, leaving winners with something like 1/4 of the listed Jackpot amount (if they choose the cash option).
In conclusion, the average payout you will receive is dependent on the Estimated Jackpot. So when the jackpot gets very high, the average payouts get closer to the cost of your ticket (and in rare circumstances may actually exceed it). But regardless of how high your average payout gets, it is cut in half if you choose the cash option and cut in half again after taxes. If you are looking for a wise investment, don’t put your money in the lottery.
Ultimately, the lottery works because of this one fact: On any given day, you can walk into a gas station and, for $1, have an actual (though extremely small) chance of writing down 6 numbers that will make you a millionaire. Just don’t count on it.
*Photo Credit: Mark Menzies (Creative Commons)