Yesterday, my coworker and I were invited to join said lottery pool. The more people who join, the less it'll cost per person. Of course, the flipside is that any potential winnings would be split between more people, so it's a bit of a wash. Apparently, last year, they won $150 and threw a little office party to celebrate. Fun, right? Well, maybe. My coworker, Rachel, didn't want to jump in without asking my advice, since I like, do numbers and stuff. It was a slow day, so I had enough time to revisit the old expected value formula with a new set of parameters.
Originally, I analyzed whether it was worth it to play the NC Powerball lottery individually. In our office lottery pool, they play both the Powerball and Mega Millions games, twice a week each, for six months. Those in the pool have to front enough money for six months of lottery tickets at the beginning of the play period, so it's a pretty steep buy-in even if it's split between a lot of people. If Rachel and I both joined the pool, there would be 17 people playing, and the price per person would be $45.88. Is a week's worth of groceries now worth the possibility of 1/17th of a jackpot later? Let's find out!
I had some trouble figuring out how the repeated play might factor in to our calculations. Obviously, the more you play, the greater your chances of winning at least once, but your probability of winning doesn't just stack up indefinitely. I thought I could use the binomial formula to figure out the probability of winning at least once, at least twice, etc. out of all 52 tickets, but each game has ten possible outcomes. Proper use of the binomial formula requires a binomial variable-- exactly two outcomes, no more no less. Technically speaking, the value produced by the expected value formula represents the sum of your winnings and losings over an infinite period of time, so the value should be accurate whether we're playing once or playing a million times.
The real sticking point for me was the $45.88 figure. Yes, the up-front cost to me is going to be $45.88, but I'm not paying $45.88 every time we play, so I can't use -$45.88 as the default "you lose" sum. I only lose that much if we never win, not once out of 52 times-- there's only about a 5% chance of that happening (chance of losing the Powerball game is about 31/32, chance of losing the Mega Millions game is about 39/40, so the chance of losing both games is about 1209/1280, and 1209/1280 to the power of 52 is just a little over 0.05). If we play the game 52 times and we each pay $45.88, then the cost per play for each of us is actually $0.88. I ended up using -$0.88 as my "lose" sum for each individual game, even though it felt wrong-- it's a lot more than eighty eight cents out of my pocket to play in the first place! But it's the closest I could get to mathematical accuracy.
Because we play two different games, and each game has ten distinct outcomes, there are one hundred possible outcomes each time we play, with one hundred different distinct probabilities. They are summarized in the chart below. (The second, darker row of numbers were just decimal versions of the probabilities without labels, for use in the Excel formulas that generated the table.)
And each of those distinct probabilities is associated with a certain payout. To calculate how much money I could take home in each situation, I added together the winnings from both games, divided by 17 players, and subtracted the $0.88 it (technically) cost me to play.
You know, I think my biggest regret is that I listed losing as its own sort of category. If I'd just given them all letters, then losing would have been "Outcome A" and winning the jackpot would have been the cleverly-named "Outcome J."
Anyway, ultimately, after multiplying each outcome's payout by its probability of occurrence and adding all those terms together, we wind up with an expected payout of -$0.79 and a decision on my part to decline participation in the office lottery pool. I sorta wanted to include all this information in my reply email, but that felt a little mean-spirited, so instead I wished the participants good luck.
It's worth pointing out that I based my calculations on the current jackpots for both Mega Millions and Powerball, but jackpots can grow or shrink over time. For fun, I set up an alternate version of the Powerball formula, and solved for the jackpot amount that would tip the scales in favor of playing (yield an expected value greater than zero). If the NC Powerball jackpot ever exceeds $287,290,619, then the day has come when lottery ticket purchases are sound financial decisions. I think they call that "Ragnarok."
Originally, I analyzed whether it was worth it to play the NC Powerball lottery individually. In our office lottery pool, they play both the Powerball and Mega Millions games, twice a week each, for six months. Those in the pool have to front enough money for six months of lottery tickets at the beginning of the play period, so it's a pretty steep buy-in even if it's split between a lot of people. If Rachel and I both joined the pool, there would be 17 people playing, and the price per person would be $45.88. Is a week's worth of groceries now worth the possibility of 1/17th of a jackpot later? Let's find out!
I had some trouble figuring out how the repeated play might factor in to our calculations. Obviously, the more you play, the greater your chances of winning at least once, but your probability of winning doesn't just stack up indefinitely. I thought I could use the binomial formula to figure out the probability of winning at least once, at least twice, etc. out of all 52 tickets, but each game has ten possible outcomes. Proper use of the binomial formula requires a binomial variable-- exactly two outcomes, no more no less. Technically speaking, the value produced by the expected value formula represents the sum of your winnings and losings over an infinite period of time, so the value should be accurate whether we're playing once or playing a million times.
The real sticking point for me was the $45.88 figure. Yes, the up-front cost to me is going to be $45.88, but I'm not paying $45.88 every time we play, so I can't use -$45.88 as the default "you lose" sum. I only lose that much if we never win, not once out of 52 times-- there's only about a 5% chance of that happening (chance of losing the Powerball game is about 31/32, chance of losing the Mega Millions game is about 39/40, so the chance of losing both games is about 1209/1280, and 1209/1280 to the power of 52 is just a little over 0.05). If we play the game 52 times and we each pay $45.88, then the cost per play for each of us is actually $0.88. I ended up using -$0.88 as my "lose" sum for each individual game, even though it felt wrong-- it's a lot more than eighty eight cents out of my pocket to play in the first place! But it's the closest I could get to mathematical accuracy.
Because we play two different games, and each game has ten distinct outcomes, there are one hundred possible outcomes each time we play, with one hundred different distinct probabilities. They are summarized in the chart below. (The second, darker row of numbers were just decimal versions of the probabilities without labels, for use in the Excel formulas that generated the table.)
And each of those distinct probabilities is associated with a certain payout. To calculate how much money I could take home in each situation, I added together the winnings from both games, divided by 17 players, and subtracted the $0.88 it (technically) cost me to play.
You know, I think my biggest regret is that I listed losing as its own sort of category. If I'd just given them all letters, then losing would have been "Outcome A" and winning the jackpot would have been the cleverly-named "Outcome J."
Anyway, ultimately, after multiplying each outcome's payout by its probability of occurrence and adding all those terms together, we wind up with an expected payout of -$0.79 and a decision on my part to decline participation in the office lottery pool. I sorta wanted to include all this information in my reply email, but that felt a little mean-spirited, so instead I wished the participants good luck.
It's worth pointing out that I based my calculations on the current jackpots for both Mega Millions and Powerball, but jackpots can grow or shrink over time. For fun, I set up an alternate version of the Powerball formula, and solved for the jackpot amount that would tip the scales in favor of playing (yield an expected value greater than zero). If the NC Powerball jackpot ever exceeds $287,290,619, then the day has come when lottery ticket purchases are sound financial decisions. I think they call that "Ragnarok."
No comments:
Post a Comment