### Math Professor & Probability Scholar on the Odds of Winning When Buying a Square at a Super Bowl Party

18-Jan-2019 4:05 AM EST

• Credit: Baldwin Wallace University

Professor Aaron Montgomery

Newswise — Aaron Montgomery is a math professor at Baldwin Wallace University in Berea, Ohio, who studies probability. He's also a self-described "sports nut" who is an excellent source of insight for those of us who don't usually gamble on the odds of winning when buying a square at a Super Bowl party or office betting pool.

For those who are thinking about buying a square for this year's February 3 Super Bowl, the odds of winning are typically better than those of buying a lottery ticket or going to a casino, Montgomery says. If you really want to win, he says, go for the scenario where you get to pick your own square(s) rather than those where you are assigned a random square.

According to Montgomery, a betting pool where you are able to choose a square means the odds are in your favor. “In the case in football square bets where that the actual numbers you get are randomly assigned, the game is essentially the moral equivalent of a slow-motion die roll,” he says. “The probability of winning any prize is just n/m, where n is the number of squares you purchased and m is the total number of squares,” he explains. For example, if you buy 25 squares on a 100-square board, you have a 25/100 = 1/4 chance of winning.

For comparison, Montgomery says, the probability of winning *any* prize on a Mega Millions ticket is about 1/24, which is roughly equivalent to buying 4 squares on a 100-square board. Probabilities in casino games such as craps and roulette vary based on the type of bet. A bet on black in roulette has probability 18/38 (about 47 percent); a craps bet on hard 8 has probability 1/36 (about 3 percent).

“The better way to think about your winnings is in terms of ‘expected values’ -- that is, the average value of each game if you played many times,” he says. “A Mega Millions has a negative expected value, meaning that if you buy many tickets, you will lose money on average,” This should make sense, he points out, because, if the operators of Mega Millions didn't make money on average, they wouldn't continue running the lottery. “The same is true of literally every lottery and every game in a casino,” Montgomery says. “If you're playing football squares by drawing squares randomly, and if the operator does NOT take a small cut of the winnings, then you're playing a ‘fair game,’ meaning the expected value is 0,” he explains.

On average, you won't make or lose money, because you don't have any advantage over the other players. “A football-squares bet is therefore better than any bet you'd find in a casino or on a lottery ticket.”

Everything changes if you actually get to pick your squares, says the Baldwin Wallace math professor. “Certain squares are much more valuable than others,” he notes. “If you get to pick your digits, aim for zeroes, sevens, and fours and stay away from fives, nines, and twos.”

The bottom line: “If you can pick your numbers wisely, you can tilt the odds in your favor and make some money on average,” Montgomery says, “especially if you're playing against people who don't do their homework about what to pick.”

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