Let me stop you right there. If you’re condsidering buying many lottery tickets for one drawing, or less lottery tickets for multiple drawings do * neither*. So

*whatever*you were thinking about doing with the answer to that question, you go right on thinking… and not doing.

But hey, you’re here for some math, right? Since I recently stole a question straight from a web forum, I decided to go fishing again. A question on Reddit’s ‘Ask Science’ subreddit found me, and I figured I’d expand on my answer to show you why, in a life or death situation where this conundrum arises… *buy all your tickets for the same lottery*. So, in the immortal words of Mr. spoolingthreads:

## Huh… What?

Well, to prove that Buyer A (and that’s our goal here) has the right strategy for Powerball you have to make a few assumptions (and by matching these up to the Reddit comments, you can even find my abridged answer).

- PowerBall doesn’t change the rules over the next year
- There will be as many drawings in the theoretical year as in 2013 (104)
- The buyer doesn’t buy duplicate tickets in the same lottery
- You’re only concerned about the jackpot, not the subprizes (the math on those are similar, but let’s only do jackpot math)

With those in mind, the problem can be worked out once you grab the odds from the PowerBall site.

First, we can calculate the odds of our ‘lump sum’ purchase, the 365 tickets for one drawing in this question, hitting the jackpot:

365 / 175,223,510

0.00020830538093889% Chance of Hitting the Jackpot

## Step Two – Odds for the ‘Dollar Cost Averager’

We’ve been doing a lot of dollar cost averaging articles on the site lately, although lottery ticket buying certainly wasn’t exactly what I had in mind. That doesn’t mean that Buyer B can’t draw any inspiration from our DCA articles – look at it like 105 separate stock price movements, although our hero puts money in daily.

So, let’s set up the odds on number two:

Averages 3.509615 tickets per drawing*

Odds of winning any one drawing are (365/104)/175,223,510, or roughly 0.000002%

Odds of not winning are (1-that last number)

Odds of not winning for a year? (The above step’s result)^104

Odds of winning a jackpot in the full year are 1 – (the above step) or

0.00020830516553883%## *We’re going to use the average, although timing matters if you were to do an exhaustive calculation. It doesn’t matter enough to swing it, though. Also, you’d need to know the drawing date.

## And There You Have It… Don’t Buy Many Lottery Tickets

In the dubious match-up of which way to least waste your money on lottery tickets, buying them all for one drawing beats buying a few for every drawing…. **0.00020830538093889%** to **0.00020830516553883%.**

Since math is definite, but not always the best way to convey a point – there are a few other ways to convince yourself of this.

One way? Pretend there are only ten numbers possible per drawing. If you buy 10 tickets, you will win 100% of the time (assuming no duplicate). If you buy 1 per drawing for 10 drawings? You now aren’t looking at an 100% chance of winning – it’s possible to lose all 10 times (In fact, the odds of losing all ten are 34.8678%!).

Bottom line – buy stock. Here, I’ll get you started – here’s a calculator on the S&P 500 back to 1871.

A little better than air powered balls, right?

save. spend. splurge. says

BF did the calculation on winning the lottery and discovered you’d need to buy either one of every number combination and shell out about $3000 a day in tickets, OR spend about 200 years (or something like that) before you will eventually win the jackpot.

Either way, it’s money down the drain.

PK says

All true – but you’re actually not guaranteed to ever win, in theory (unless we start playing with infinities – but longer than any human will live, haha).

Jason says

If the question is about maximizing chances of winning once, the math is correct. However, the expected value is the same for both scenarios since even though the second scenario has a lower chance of winning once, it can win multiple times unlike the first.

Either way, don’t play.

PK says

I thought about bringing that up, but you’d have to introduce another assumption – that the prize amount doesn’t change. Technically, since the Powerball prize changes based on whether the previous drawing was a win (and on the number of tickets sold), you could do that math by introducing some Markov – but it just seemed easier to leave the whole thing out, haha.

I did actually try to run that math on the Mega Millions if you’re interested – this calculator can predict ticket sales by purse amount, for what it’s worth: http://yourdayjob.net/mega-millions-expected-value-calculator/

AvgJoeMoney says

That meteor last week was more likely to hit the earth than you are to win the lottery. “What meteor?” you ask? Exactly….. (You know it’s horrible odds when I have to look 10 places right of the decimal point to find a difference…..)

PK says

“You can’t win if you don’t play!”

I don’t know, I can name plenty of millionaire who got that way without playing the lottery. (Maybe they can clarify “win”?)

Clarisse @ Make Money Your Way says

I’m not a fan of buying lottery tickets, but one of my friends is very addicted about lotto. She did the calculation and buy lots of lottery tickets, sometimes she won, but I think she just wasted her money in doing that.

PK says

Almost undoubtedly – but since we’re a bit math obsessed on this site, we can come up with a few examples where that wasn’t the case:

http://yourdayjob.net/beating-the-lottery/