Here’s a mental challenge: Let’s say I’ve chosen a particular second in time over the past nine years. Between November 2014 and today, I’m thinking of a specific (and totally random) year, month, day, hour, minute, and second. Could you guess it? No possibility? You have a better chance of guessing a specific second in a nine-year span than you do of winning the Powerball.
Last months power ball made headlines for winning a colossal $1.7 billion jackpot, the second largest in the history of the game (the winner had not claimed his prize). everyone knows that your chances of winning the lottery is scarcer than scarce. But when jackpots accumulate to record-sized prizes, could the massive payout potential ever outweigh the rarity of winning? In other words, is the lottery ever a good bet? The answer may surprise you, when even a good bet It might turn out to be a bad idea, mathematically.
Mathematicians sometimes separate good bets from bad bets using a concept called expected value. Consider the example of betting on the outcome of a dice roll. It costs $1 to choose a number between one and six. If you guess the spin correctly, you win $1 and if you guess incorrectly, you lose your dollar. Would you accept that bet? It seems unfair because you can win exactly as much as you lose ($1), but you are much more likely to lose (five out of six spins lose).
What if it only cost you $1 to play, but you would win $100 if you guess correctly? Suddenly, the prize seems big enough to outweigh the probability of losing. Some probabilistic reasoning can tell us exactly what cutoff value should make one tempted to play and not disdainful.
Clearly the relevant variables are: how much it costs to play, how much you can win and the probability of winning. The expected value of a bet is converted to a weighted average where the possible outcomes (wins and losses) are weighted according to the probability of each occurrence:
Expected value of a bet = (probability of winning) x (amount won) – (probability of losing) x (amount lost)
The solution to this equation reveals how much money you could expect to win (or lose if it is a negative number) per bet in the long run if you made the bet many times. For example, with our dollar bet on the result of a die rollthe probability of winning is ⅙, while the probability of losing is ⅚, and we can lose or win $1.
Expected value = (⅙) x ($1) – (⅚) x ($1) = –.667
If we made this bet many times, in the long run we would expect to lose around 67 cents per bet on average. A similar calculation with the payout of $100 yields an expected value of almost $16, clearly a good bet. This framework also allows us to calculate a payoff in which the bet is perfectly matched, where the long-term expected value is $0. For a die roll, this equilibrium payoff reaches $5 because you are five times more likely to lose than to win; so a reward five times the cost offsets the risk.
Let’s apply the expected value lens to Powerball. The jackpot starts at about $20 million and a ticket only costs $2. The probability of winning the jackpot: one in 292,201,338. If we look at these numbers, the lottery ticket will have an expected value of approximately –$1.93. You would get more value from those two dollars if you exchanged them for ten cents.
This calculation ignores several subtleties for the sake of simplicity. For one, it assumes that you choose the annuity option, which spreads your earnings in annual installments over 29 years instead of a cash lump sum (the annuity is worth more in the long run). Second, taxes ensure that you will never leave with a full wallet. Winning big would put you in the highest tax bracket, so 37 percent of your windfall would end up going to Uncle Sam (this doesn’t include state taxes, which vary by state). Powerball also awards smaller prizes for partial matches of the numbers drawn, while we have only considered the jackpot. There is another important consideration that I have omitted and which I will discuss below. But factoring all of these details into your calculations will only make ($1.93 seem generous) the bill actually worth even less.
Still, a $20 million jackpot pales in comparison to last month’s $1.7 billion. If no one wins the jackpot, the total prize pool rolls over to the next drawing. When the pool continues to grow for many consecutive weeks, there is surely a point at which the overwhelming prize overshadows the minuscule chance of winning, much as the $100 reward on the dice outweighs the mere one in six chance of guessing it. After all, the probability of matching all six numbers does not change and the cost of the ticket does not increase. It turns out that not only are huge jackpots often still bad bets, but, paradoxically, they also tend to be worse bets.
A multimillion-dollar payout appears to offset the probability of winning by approximately one in 300 million to generate a positive expected value for a ticket. In fact, this statement is often circulated in the media around mega jackpots. But it overlooks a crucial detail: multiple people could win the jackpot and therefore split the winnings. We need to add more terms to the expected value calculation to take into account all possible outcomes, for example (probability of keeping the only winning ticket) x (jackpot) + (probability of splitting the jackpot with another ticket) x (half of the jackpot), etc.
We have established that winning the lottery requires overcoming super low odds. Wouldn’t that mean two winners in the same draw silly low odds? Sometimes, but when hundreds of millions of tickets are sold, collisions are more likely. For example, the first jackpot to reach $1 billion occurred in 2016, exceeding $1.56 billion. The hype surrounding the new album sparked a buying frenzy and more than 635 million tickets were sold. (That’s more than 20 times the number of tickets sold in an average Powerball drawing that year.) With so many tickets in circulation, the probability of more than one winner exceeded 60 percent! In fact, three winners ended up splitting the grand prize in 2016. When taking into account the total number of players, tax withholdings, and secondary prizes for partial matches, even this gigantic jackpot did not offer positive expected value. We omitted the jackpot split detail in our calculation of the expected value of the $20 million Powerball above because smaller jackpots attract smaller crowds and have a slimmer chance of splitting. Plus, with a negative expected value of $1.93, we hardly needed another factor to convince us it was a bad bet.
Side note: The 60 percent figure assumes that the ticket numbers are chosen at random, which is not exactly the case. Although all sequences of six lottery numbers have an equal chance of winning, many people choose their numbers personally and tend to choose sequences that mean something to them, such as birthdays or anniversaries (resulting in many numbers under 31). People also seem to prefer odd numbers and numbers that are not multiples of 10, perhaps because they seem more random. This behavior increases the chance of splitting the pot in draws with smaller random numbers, but decreases it in other draws. So while you can’t increase your chances of your numbers coming up, you can decrease your chances of splitting the jackpot by choosing large even numbers and including multiples of 10.
Buying manias have subsided since 2016. In fact, the two largest jackpots in lottery history (last month and last year) attracted so few buyers that the expected value of a ticket tipped into positive territory. , even after adjusting for caveats like taxes and profits. -terrible. Sometimes lotteries offer what we call a “good bet” here. Smaller state lotteries might even be better places to look for positive expected value, since they tend to generate less advertising and sell fewer tickets.
Don’t empty your emergency fund at the nearest convenience store just yet. Despite admitting that the expected value of a ticket can sometimes seem attractive, I’m going to backtrack and explain why I still think the lottery is a bad bet.
Lotteries with positive expected value are rare. And, more importantly, you probably won’t be able to identify it in time to place a bet, because ticket sales figures are not published before the draws. As we have seen, larger prizes do not necessarily mean a higher expected value. So while occasional lotteries offer a good bet, predicting which The lottery is a game of chance in itself. Even if you could identify them, expected value may not actually be the best indicator of a “good bet.” Expected value is useful for medium-sized problems, such as a $100 die roll, but may not adequately capture all relevant considerations in extreme situations such as lotteries. It turns out that sometimes even a good bet is a bad idea.
On the one hand, the expected value is based on long-term behavior. You don’t actually expect to win $16 when you bet on our $100 spin. In fact, you can’t win $16; you will lose $1 or win $100. The $16 is what you would expect to win per bet on average if you keep playing repeatedly. Lottery wins are so rare that this long-term average can never realistically be achieved. Second, money loses value as you continue to accumulate more. Your second $50 million won’t bring you as much joy as your first $50 million. Expected value analysis treats each dollar equally and does not account for diminishing marginal returns. Relatedly, expected value ignores personal risk aversion. People tend to dislike losing money more than winning it. So, while expected value is excellent for mathematical evaluations of probabilistic systems, it does not fully model human psychology and decision making.
Now, to backtrack on my backtrack: a lottery ticket costs $2. Players don’t buy an investment, they buy permission to fantasize for a couple of days. We all make frivolous purchases and most of them have zero chance of making us a fortune. Money spent on lottery tickets doesn’t end up thrown into the ocean either. Much of the revenue funds public services such as education. There has even been some research suggesting that anticipation when playing the game makes people happy regardless of the result. So while I can’t recommend playing the lottery on a mathematical basis, there’s a lot more to a happy life than math. Or so I’ve been told.
This is an article of opinion and analysis, and the opinions expressed by the author(s) are not necessarily those of American scientist.
