A good bet? W&M professor on the math and psychology of Powerball
Harvey Langholtz, a professor of psychology at William & Mary who teaches a class on decision theory, recently talked with W&M News about what people should consider when deciding whether or not to play the Powerball lottery. – Ed.
How does Powerball work?
A Powerball ticket costs $2. To play you then select numbers corresponding to five white balls numbered 1 to 69, plus one red Powerball number between 1 and 26. When the game is played every Saturday and Wednesday evening, a machine randomly picks five white balls and one red ball. To win the jackpot, your selected numbers have to correspond exactly to the numbers the machine selects. The size of the jackpot can vary. If there is no winner on any given Saturday or Wednesday, the amount is carried over to the next game.
What’s the probability of winning, and how is that calculated?
As stated on the Powerball website, chances of winning are one in 292 million. Here’s the math behind that:
The likelihood of someone correctly guessing the first of the five white balls is 5 (because there are five numbers that will turn out to be right) out of 69, or 5/69. The likelihood of guessing the second, given that the player has correctly guessed the first, is 4 (only 4 remaining that will turn out to be correct come the time of the drawing) out of 68 (because you already eliminated one of the original 69). So that works out to 4/68. And the likelihood of then correctly guessing white balls three, four and five, are 3/67, 2/66, and 1/65. And the likelihood of a player guessing the correct number on the one red Powerball is 1/26. So what is the probability of guessing all five white ball numbers and the red Powerball number? It is 5/69 x 4/68 x 3/67 x 2/66 x 1/65 x 1/26 or 1/292,201,338.
There are smaller payouts for partial correct guesses. These can also be calculated but for the sake of the present analysis, I have ignored these. It gets complicated.
From the standpoint of Decision Theory, how would a player go about deciding to play?
It gets down to the concept of Expected Value or EV. The EV of any event can be calculated as the sum of the products of the probability of each possible outcome and the value of each outcome, or EV = ∑(Pi x Vi) where ∑ is the summation, Pi is the probability of each event and Vi is the value of each event.
An example of this I use to teach the concept in my classes on Decision Theory at William & Mary is one of flipping a coin and paying the participant a dollar if it shows heads but nothing if it shows tails. It’s clear that 50 percent of the time it will show heads and you will win one dollar and 50 percent of the time it will show tails and you will win nothing. So what is the expected value of this event? It is EV = ∑(Pi x Vi) or EV = (.5 x $1.00) + (.5 x $.00); EV = $.50 + $.00; EV= $.50. So the Expected Value of this bet is fifty cents. Will you ever win 50 cents on any one play? No. Half the time you will win a dollar, and half the time you will win nothing. The average of these, or the EV, is 50 cents each time you play.
In this example, how might Expected Value impact your decision to play?
First, a potential player has to decide what they’d be willing to pay each time to play this game. There is really no right or wrong answer. If the player is willing to pay 50 cents each time to play this game, that’s a fair bet. The player would not have an advantage or a disadvantage. They’d be paying fifty cents for something worth fifty cents. If someone played this game all evening at the end of the time they might be ahead or behind by a few dollars due to chance but the likelihood of gaining or losing a large amount would be very small.
What if they player answered 49 cents or 48 cents or 25 cents or anything less than 50 cents? They would be telling me they’d be willing to pay but only if there were an advantage to themselves. If someone actually paid 25 cents each time for something worth 50 cents they’d average a gain of 25 cents each time.
If the player answered 51 cents or 52 cents or 75 cents or anything over 50 cents, they’d be indicating a willingness to pay a premium for the opportunity to play the game. If someone were to pay 75 cents per play they would, on average lose 25 cents each time. If they played the game 100 times they’d probably be down about $25.
At this point in Decision Theory class, I ask my students if people are ever willing to play a game where they pay more than the Expected Value. They realize quickly that this is exactly what is going on with every game in Atlantic City or Las Vegas – slot machines, poker, black jack, roulette, or any other. People are there right now paying more than EV. And people pay more than EV when they play Powerball most of the time. But maybe not in the case of the Jan. 13 drawing.
How would you calculate EV for Powerball?
If a player’s chances of winning are one in 292 million, as in Powerball, and if you are paying $2 for a ticket, the amount of the Powerball payout would need to be 2 x 292,201,338, or $584,402,676 for you to be getting an even bet. Remember the game of flipping a coin and getting one dollar for each head that I described above? In that game, the likelihood of winning was .5 and the payout was $1 to make the EV worth the 50 cents you would pay. In the case of Powerball, the payout needs to be $584,402,676 to make your wager of $2 worth $2. But in this case, it gets more complicated. Will you really be able to keep the full $584,402,676? Hint, the answer is no, half will go to taxes. And what if one or more others also guess the same numbers as you do? The jackpot would then be divided between or among you.
With Wednesday’s Jackpot estimated at $1.4 billion, is it now a good bet for you?
It depends. There are two ways the winnings could be taken. One is $868 million in cash. Taxes would take about half of that, leaving about $434 million. That’s less than the $584,402,676 you need to make your $2 ticket worth $2 (see explanation above). That’s a losing bet. So you might ask about the option of $1.4 billion doled out in annual payments for 30 years. Half would still go to taxes. And do you expect to live for another 30 years? What happens after that?
The point of all this is that while the size of the current jackpot sounds so tempting, it’s still a struggle to justify the cost. It might be close to a break-even proposition, but even that’s questionable. We should also take a moment and consider what the advertised minimum is for a normal Powerball jackpot where there has been no carryover from the previous drawing. It’s $40 million as an annuity. That’s far below the current $1.4 billion. So if you play the normal Powerball game with a jackpot of $40 million, the $2 ticket you purchased will have an expected value of $40,000,000/292,201,338, or about 13 cents.
I once heard the joke of the man who went to Las Vegas to gamble and before he left he said, “I hope I break even. I need the money.” So in the case of this Powerball game, if you need the money, perhaps you should keep it. Put it in the bank.
With an understanding of the math and knowing that the odds of winning are one in 292 million, is there anything wrong with an individual deciding to place a $2 bet?
Wrong? No. It’s a personal decision that is all about understanding the probability and calculating the personal risk. However, there are some who raise serious questions about the demographics of lotteries, the populations from which lottery revenues are raised, and whether the state is teaching – and sponsoring – the intended set of skills and behaviors needed by society. In a 2011 article published by V. Ariyabuddhiphongs in the Journal of Gambling Studies, the author concluded that the “poor are still the leading patron of the lottery … The legalization of gambling has seen a significant increase of young people gambling, particularly in lotteries, and the best predictor of their lottery gambling is their parents’ lottery participation.”
A 2010 paper by Wiggins, Nower, Mayers, and Peterson in the Journal of Community Psychology titled “A geospatial statistical analysis of the density of lottery outlets within ethnically concentrated neighborhoods” found that lottery ticket outlets are more common in minority neighborhoods and contribute to gambling addictions
A 2012 study published in the Journal of Gambling Studies by Barnes, Welte, Tidwell, and Hoffman, titled “Gambling on the Lottery: Sociodemographic Correlates Across the Lifespan” found that those in societies’ lowest 20 percent in terms of socioeconomic status had the “highest rate of lottery gambling” and concluded that “increased levels of lottery play are linked with certain subgroups in the U.S. population – males, blacks, Native Americans and those who live in disadvantaged neighborhoods.”
Of course none of these serious social, demographic, political and even ethical issues get any of the coverage that the lotteries themselves do. In a free society, we are all at liberty to spend our money as we wish. But the social scientists who question the role of the state in promoting gambling – which was previously conducted by organized crime until the states took it over – make a valid point that is worthy of our consideration.
So, professor, do you think you will purchase a Powerball ticket this time in advance of the drawing on Wednesday night?
I may stop by my local convenience store on my way home from campus this afternoon and purchase a ticket in the hope of getting a piece of this big jackpot. But I’ll buy only one ticket and only this one time. If I win I’ll set the money aside to pay for my grandchildren’s education, and I’ll make significant donations to charities that serve the most needy and vulnerable in society. But the rest of the year I won’t be buying Powerball tickets. I’ll be saving for my retirement and for my grandchildren’s education. I need the money. They will too. And of course the most important things in life are free anyway.
