Speaking of Lotto and Probability

Remembering the conversation from Probability and MegaMillions, here’s how to rig the game. Buyer beware, this has been tried….

Probability of winning the MegaMillions lotto: 1 in 176Million

Sure BetThe probability of winning and everyone else losing does not change much as you add more people to the mix because the probability of losing is so close to 1 or 100%. Raising the loser multiplier (.999999994) to a power doesn’t change the group multiplier much. For the MegaMillions you don’t have much incentive to keep people out of the game. Unless . . .

you rig the game….


Mathematically it would make sense to buy one of every combination if the prize is over $176Million, right?

Now you have to be much more concerned about competition. (Plus, last I checked this is illegal and it is practically impossible, imagine if you forgot ONE tiny combination….) If you buy all combinations, you will win. Period. Buy one of each, you will win. You will also spend $175,711,536 in after tax dollars for the privilege of winning. How big does the pot need to be in order to justify spending almost $176M? (and how many man-hours would you need and how much would you have to pay your army of buyers…. There’s so much great math here!!)

At 35% tax rate (federal and state), you need to have prize money of $270,325,440 just to break even, without accounting for time value of money. Oh my.

Now consider that these lotteries pay out over 20-26 years or you get one lump sum. The lump sum is some percent of the total. Simplifying, the 20 year option grants you the original amount of your win, but divides the cash into 20 equal payments without earning interest on the principal they have tied up.

Assume lump sum payout of 71.2%: $244,451,219 BT

Assume your annuity discount rate is 4% (what you could earn now on that money if you had it in your hands, t-bills): $258,583,247 BT

Now let’s tack on those pesky taxes….
Lump Sum: $376,078,798 payout necessary to breakeven today.
Annuity over 20 years: $385,005,613 to break even at the end of 20 years.

So now essentially the prize has to be over $380Million to justify spending $176Million, more than twice the cash outlay. What would our return be if the jackpot rose to $1Billion?

One Billion Dollars

onebilliondollarsLump Sum
We would take home $467,220,000, 166% return as long as we didn’t have to share. If one additional person hits the jackpot, our total falls to $233,661,000, a 33% return. If more than 2 people happened to pull the right combinations . . . we would lose buckets of cash. At simply three winning tickets, the pot is down to $155,740,000 or an 11% LOSS.

The annuity would pay out $650,000,000 over 20 years, and I would value that at $441,685,606 based on our 4% discount rate, a total gain of 151%. Again, pesky winning ticket holders would reduce our draw substantially. If we had to share with one other person: $220,842,803, return of 26%. And heaven forbid three people win, we are down to $147,228,535 a staggering loss of 16%. Bad day indeed.

This is where the study of risk gets interesting. While the probability of anyone else having that same ticket is minuscule, the ramifications of a second ticket holder are huge. That’s part of what keeps the lotto market in check. Some would argue the legality of buying one of each keeps the market in check . . . I would argue legality never keeps a smart gambler at bay. If there’s money to be squeezed, someone will figure out how to do it.

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