Why Any Smart Investor Should Play The Powerball. Sort of!
If you’ve ever driven on any interstate roadway or up our very own west-side highway toward Washington Heights, you could not have missed the massive illuminated advertisement signs: “Current Powerball _______ million, Current Mega Millions ______ Million.” As a driver who just passed these lit up fantasies of hope, you’ll need to ask yourself the million-dollar question, “Should I buy a ticket?”
To answer this question, you’ll need to determine a lottery ticket’s expected value by evaluating it from an economic and mathematical perspective.
The expected value of any investment is calculated by simply multiplying each outcome by the probability of a specific outcome occurring. For example, the expected value of a coin toss that offers $10 for a “heads” and nothing for a “tails” is $5 dollars ($10 x 50% plus $0 x 50% = $5). If the cost of entry into the coin toss gamble is $4 dollars, then it is economically sensible to play. By contrast, if the cost is $6 dollars, the expected value is less than the entrance fee and it is not a sound investment.
In regards to the Powerball, does the expected value of the Jackpot drawing exceed its ticket price? First, we’ll need to consider all of the nine different ways that you can receive a payoff from the lottery. The Powerball consist of five white balls ranging from one to 69 and a red ball ranging from one to 26. Thus, the total expected value would be the probability of winning each prize multiplied by the amount won at that level. For example, the chance of matching one red ball (and only one red ball) is 1 in 38.32 and the payout is $4. The expected value of this prize is approximately 10 cents. ($4 multiplied by 1/38). The sum or total expected value for the first eight prizes combined (excluding the jackpot) computes to approximately 26 cents. So at the ticket price of $2, it is clear that you should not purchase a ticket. (To understand the calculations feel free to visit http://www.flalottery.com/exptkt/pwrball-odds.pdf).
But of course, you don’t play the lottery to come in one of the eight runner-up positions. To win the entire Jackpot, however, you must match all six balls which, according to the Multi-State Lottery Association, only occurs once in every 292 million. Further, even if you are lucky enough to match all 6 numbers, your jackpot winnings still vary from drawing to drawing. Therefore, according to the original expected value, you would need the entire jackpot to be 584 million. You’ll arrive at this figure after determining the expected value of 584 million multiplied by a 1/292 million chance of winning is = 2 which is the break-even point on your original $2 investment (the ticket price).
But to truly make an informed decision, you’ll need to also consider the post-tax or “net” earnings; The Jackpot figure that is displayed on these interstate signs isn’t actually what a lucky winner takes home. The cash value of the jackpot is usually 60% of the number that the signs display, from which taxes must still be deducted. After taxes you will be left with somewhere between 33%-40% (dependent on the respective state tax rate) of the number that is displayed on the signs. (New York has the highest state tax whereas several states have no state tax on lottery prizes).
How do these these adjusted figures affect the expected value? If you assume that the final cash payout needs to be 584 million to break even, and that cash payout is somewhere between 33%-40% of the displayed sign, you would actually need the signs to display a figure of approximately 1.6 billion dollars (assuming a take home over of about 36.5 percent- the average of 33 and 40) in order for your expected value to break even on your original investment. Such a number has only occurred once in Powerball history, on January 13, 2018.
So, if the Powerball were to reach 1.6 billion, should you then dump your retirement savings and play the Powerball? After all, according to the above estimates, it would be worth it, right?
Well not really!
The prudence of this investment hinges on there being only one winner to claim the entire jackpot. However, the way the jackpot works is that if there are multiple winners, they all split the jackpot evenly. So in your situation, the 584 million indifferent price becomes 292 million (584/2) and thus the expected value would drop to $1 which is below our ticket price. Therefore, you wouldn’t even contemplate buying a ticket!
But should you necessarily assume that there would be 2 winners? After all, you were lucky enough to win the lottery; maybe you’re also lucky enough to be the only one?
To determine the probability that there will be more than one winner, you’ll need to return to the January 13 example. There, 1 billion tickets were sold the night before the drawing, allowing for the probability of three winners (1 billion tickets, divided by 1 in 292 million chances) who then split the jackpot evenly. Under those circumstances, the expected value of a ticket drops below the $2 threshold. Indeed, that’s exactly what happened in the the January 13th drawing; three individuals received over 200 million dollars a piece.
Let’s rap up this discussion and give a concrete answer. If you are a rational investor, then the answer is overwhelmingly NO! Economic theory teaches that people are rational decision makers; If you prefer pizza to pasta, and prefers pasta to lasagna, then you’ll also prefer pizza to lasagna. So why should you play? Because it’s also true that people don’t always abide by rules of rational decision making. After all, if you had pizza Monday and Tuesday, and then it’s offered again Wednesday you might want a little taste of some lasagna. Personally, my sentiment is that people aren’t rational thinkers and the expected utility (or satisfaction) you receive from playing, praying, and hoping that it’s you who will become the millionaire, trumps the pure mathematical expected value, which would otherwise never allow you to rationally buy a ticket.
So is my entire article a waste? No, not at all! You should just know that mathematically it never makes sense to play the lottery but so many people do because as the lottery so brilliantly advertises, “Hey, you just never know!”