mega millions math
by santoki
From the roof deck bar atop the Hotel Gansevoort Tuesday past, the Imperialist, his cronies and I entered into a pact. It was similar to that which took place across the country – we would share the Mega Millions jackpot. Mind you, when we entered into the pact, the drawing had already taken place. You see, I bought $10 worth of tickets. Since we were all unaware of the results, they took the opportunity to buy into the potentially valuable tickets. Kind of like taping the game. Until the outcome enter your reality, anything is still possible.Everyone chipped in $2.50, and we were all proud shareholders in a fortune.
We won! Well, okay. The payout was only $3. Okay, according to NY State law, 2 out of the 3 Brits were ineligible to play. That’s neither here nor there.
Here is the ha ha. An interesting thing happens when smart people gamble. They tend to toss logic off the roof deck. Seriously, you wouldn’t believe the cock and bull math swirling about. Odds of winning were halved when doing this, or quartered when doing that. Now I don’t claim to know a ding dong thing about math, but I am pretty sure that in this case, my fu is good.
Here’s what we do know:
When buying a ticket for the Mega Millions, you will pick 5 numbers from 1 through 56. Next, you will select one number from 1 through 46. At the time of the drawing, if your 5 numbers match the numbers on the white pingpongs, and your solo number matches the drawn super duper deluxe gold mega ball, you are the mega winner. Doing a little mega math, we find that the odds of winning the mega prize are 1 in 175,711,536.
Combinatorics, baby!
So what happens when you buy 2 tickets? According to some of our lotto-drunk cohorts, the odds are cut to 1 in around 88 million. By that math, you are either cutting your odds in half with each ticket you buy, meaning you only need about 28 tickets to hit the big one, or you might as well stop after 100 tickets, as the rate of improvement with each dollar you spend will be a fraction of a percent.
Frankly, I don’t buy into any of this cockamamie hocus pocus. That is because for all of those numbers, there are 175,711,536 possible combinations. The winning outcome represents 0.000000569114597006311% of the total, a mere drop in the pool of combinations. If this was math class, we would round to zero. If you buy 2 tickets, you are selecting 2 possible combinations out of 175,711,636, or 0.00000113822919401262% of the total. Again, we would usually round to zero. And 20 tickets? Your chance of hitting the jackpot is a just over a hundred-thousandth of a percent. Say it with me, round to zero.
With all that, I say happily that girlfriend will keep buying those $1 day dreams, even though the math might say that there is zero chance of winning.
So there.
cunctabundus 
Your friends were right on two tickets doubling your odds over one ticket.
Really? Huh.
Here’s how the math works:
Say you figure out the single ticket odds, in this case about 1 in 80M. As you increase the number of (distinct) tickets, you increase the number of possibilities that win out of the total number of tickets. So if you buy 100 tickets, your chances are 100 in 80M. Still not great, but better than before.
The real question (for math nerds) is whether the ticket is fairly priced. In other words, if you bought all combinations and won, at what ticket price would the money you put out be equal to the money you win? That’s called the Expected Value. If you’re paying less than the Expected Value, you’re at least getting a “fair” deal. And you know that the house isn’t pocketing an excessive amount of the investors’ money.
There are cases in which the Expected Value is acceptably much less than the price of a ticket. In a Raffle, for example. In those cases, the buyer recognizes the difference and assumes that the extra cost supports some worthy cause.
By the way, I found your site because it occurred to me that in these days of instant web collaboration, there ought to be someone who would be discussing the Weekend Edition Sunday Puzzle. I’ve gotten answers for the last month or so without much trouble, but the football stuff is pretty hard for me :-).
Thanks Ted! Someday, I hope to be a math nerd. Or at least a professional gambler. Either way the 411 will come in handy, I am sure.
I took a tiny break from the blogging, so I am a little behind on the WESP. Time to get back on it, I guess. I usually post the answer about 5 minutes before the deadline, though I’m sure that you’ll have it figured out by then!
What if one had $175,711,536.00 and bought that many tickets when the mega millions was say $250,000,000.00 ? Only better hope you’re the only winner.
I love the odds. Not so much the ridiculous odds, but the fact there ARE odds of winning to begin with. I love it.
Gill, It’s been tried. It’s no small matter to get 175 Million tickets printed in a few days, and the Mega-Millions folks won’t just give you a blanket pass, you have to physically buy the tickets.
If you try to save time by getting 175M autopicks, the odds of you drawing multiple tickets with the exact same numbers on them goes up. While the odds of any two particular tickets matching are also 175 million to one, there are billions of combinations of 2 tickets taken from a set of 175M. So you are almost certain to have a hole in your odds of winning.
I was always curious of the formula.