Let me stop you right there. If you're considering buying many lottery tickets for one drawing, or less lottery tickets for multiple drawings do neither. Whatever you were thinking about doing with the answer to this question, you go right on thinking... and not doing.
But you're really here for some math on the optimal number of lottery ticket to purchase, right? If you're going to buy and we can't convince you not to, we'll run the numbers.
A question on Reddit's 'Ask Science' subreddit inspired this question. In a theoretical life or death situation where this conundrum arises... buy all your tickets for the same lottery drawing.
In the immortal words of Mr. spoolingthreads:
The Right Strategy for Buying Lottery Tickets
To prove that Buyer A has the right strategy for a drawing like Powerball you have to make a few assumptions:
- Powerball doesn't change the rules over the next year
- There will be equally many drawings in future years
- The buyer doesn't buy duplicate tickets in the same lottery drawing
- You're only concerned about the jackpot, not the sub-prizes (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 a 'lump sum' purchase, 365 tickets for one drawing in this question. Here are the odds to hit a jackpot:
365 / 292,201,338.0
0.000124913870175% Chance of Hitting the Jackpot
Odds for a Lottery 'Dollar Cost Averager'
The Dollar Cost Averaging analogy means a person would spread the same number of purchases out over other drawings.
Let's set up the odds on number two:
Averages 3.509615* tickets per drawing (based on 104 drawings a year)
Odds of winning any one drawing are (365/104)/292,201,338.0, or ~0.0000012010949055%
Odds of not winning are (1-that last number)
The 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
*We're going to use the average, although timing matters in an exhaustive calculation. It doesn't matter enough to swing it, though. Also, you'd need to know the drawing dates...
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.... 0.000124913870175% to 0.000124913792665%.
Math gives us precision, but it's rarely the best way to convey a point. There are a few other ways to convince yourself of this.
Here's a good way:
- Pretend there are only ten possible numbers per drawing.
- If you buy 10 tickets, you will win 100% of the time (assuming no duplicates).
- If you buy 1 per drawing for 10 drawings, it's now possible to lose all 10 times (In fact, the odds of losing all ten are 34.8678%!).
The bottom line - don't buy multiple lottery tickets. Buy stock. Here, I'll get you started - here's a calculator on the S&P 500 back to 1871.
Better odds than air powered balls, right?