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Lottery Math

Updated on September 2, 2012

Lottery Math

Let's look at some basic lottery math. As you've no doubt heard many times "The Lottery is a tax on people who are bad at math." Perhaps we can correct a few mathematical deficiencies in order to clarify elementary lottery calculations.

Lottery Math

Let's start simply. Imagine that somewhere in the world a 1 number lottery exists. Players plunk down a single dollar and get to pick a single number. For the sake of argument our imaginary number runs from 1 to 100.

What are the odds of winning? That depends on what you consider "winning". The statistical odds of picking the correct number are 1 in 100. That can also be stated as 1 to 100 against. In other words, if you play the game 100 times, statistical analysis indicates you will win one time. Statistical analysis only goes so far; you might win 100 times in a row, but that would be very improbable. 100 consecutive wins is not impossible, just extremely unlikely.

Now, the very smart players (depending on how you define 'smart') will quickly realize that purchasing more tickets increases the odds of winning. The second ticket purchased actually reduces the odds from 100:1 to 100:2. The odds get reduced. Your chances of winning instantly become twice as good. Given that all 100 possible numbers can be covered by only $100 dollars worth of tickets, savvy players soon figure out that a relatively small investment can guarantee a winning bet. Actually, forking over $100 for all 100 possible numbers morphs the lottery game into a sure-thing. Anyone willing to plunk down a C-note is guaranteed a winning ticket. The only question becomes whether or not the investment will turn a profit. A $101 dollar pay out is necessary to make this strategy worthwhile. Receiving $101 dollars for each $100 spent is a 1% return on investment, or ROI. Doing the same thing every week (assuming the our imaginary lottery is a weekly event) would realize a 52% ROI over a calendar year. At the end of the year you'd have $152 in your pocket.

For better or worse, lottery players are competing (so to speak) with each other. In order for this scenario to blossom, we need 101 other players to not pick the winning number and no other players to pick the winning number. This admittedly contrived scenario would provide us with every other players money and eliminate the obligation to split our winnings with any other player.

The ultimate payout for each drawing is a function of the number of players who risked their money. If 10 players hit the winning number, the payout must be divided equally amongst all 10 players. Referring to the above example, our 1% payout plummets to 0.1%. We make 10 cents a week.

Hopefully you've realized that in short order every player would be playing every number. Every player would win every drawing and the winnings would become diluted to the point that playing the game would hold no fantasy.

More Lottery Math

Our next lottery math scenario will be slightly more real-world. Let us assume a 3 number game, often referred to as a "Pick 3". The numbers range from 1 to 50. The odds of picking any single number is 1 in 50, or 50:1 against. The odds of picking all three numbers are calculated as the product of the individual probabilities. It ciphers out to 1:50*50*50, or 1:503, or 1:125,000. Building on our previous example, you could theoretically purchase all 125000 winning number combinations and be guaranteed a winning ticket. Given that the Pick 3 payout is typically much higher than $125,000 dollars, this scenario appears financially viable. Unfortunately, standing in line at the gas station 125,000 times becomes problematic. Is there a middle ground? Consider purchasing 5 tickets instead of a single ticket. The second ticket reduces your odds slightly. The third ticket reduces your new odds again. Refer to the following table for a detailed breakdown of the odds.

Odds are not cumulative accross multiple games

Each week the odds reset. In other words, your odds of winning don't improve simply because you played last week or have played every week since 1942. If you miss a week, your odds don't get worse the next week. The individual weekly plays are not functionally related or interdependent in any way, therefore the casual player who waits until the jackpot reached 100 million has the same chance to win as the 'faithful' player who buys the same numbers every week without fail.


  • None of our payout calculations include profits skimmed by the institution running the game.
  • Play at your own risk.
  • Don't expect to win.
  • Take a course in statistics, then reconsider your allocation of entertainment dollars.


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    • profile image

      Marten 4 years ago

      You write:

      " Let us assume a 3 number game, often referred to as a "Pick 3". The numbers range from 1 to 50. The odds of picking any single number is 1 in 50, or 50:1 against. The odds of picking all three numbers are calculated as the product of the individual probabilities. It ciphers out to 1:50*50*50, or 1:50^3, or 1:125,000."

      What you left out is that this probability is only for a game in which you pick 50 numbers that are not necessarily distinct, and where the order in which you pick them matters.

      But there are also Pick-whatever games where you have to pick all different numbers, and where the order doesn't matter. I believe these are more common, for instance in my area we have a Pick 5 and you choose 5 unequal numbers out of 35.

      If the Pick-3 game required that the numbers be different and the order was not considered, then the number of possibilities would be 50x49x48/6 or 19,600. Far less than 125,000. This is because you have 50 ways of choosing the first number, but only 49 ways to choose the second number because one has already been removed, and then 48 ways to choose the third number because 2 have been removed. As there are 6 ways to arrange 3 things, you divide by 6.

    • sneaxy profile image

      sneaxy 6 years ago from United Kingdom

      I think there were two things left out here:

      Firstly, people (arguably) play lottery for the excitement and thrill of seeing their numbers come up in the draw.

      Secondly, if you calculate the expected return on a £1 lotto ticket with the odds mentioned in the article, you do get that 52% expected return that you mentioned, the question is, where's the remaining 48%?

      We used a very similar example in our finance class to work out the expected ROI on a lottery ticket and to cut a long story short we came to the conclusion that like @The Old Firm mentioned, you should think about lottery tickets in the same way you buy a beer: you're paying 48p for the opportunity of being thrilled (and disappointed, thus an emotional roller coaster) and the rest is just an investment with a negative return.

      If it's the thrill you're after, then try this, pick your six numbers for this week's draw. Write them down on a piece of paper and DON'T buy a lottery ticket. Keep those numbers with you and then sit in front of the TV when the draw is taking place. If you're six numbers come up, that will be the thrill of your life! lol...

    • nicomp profile image

      nicomp really 6 years ago from Ohio, USA

      Just noticed the bad grammar in the first sentence. How embarrassing.

    • nicomp profile image

      nicomp really 7 years ago from Ohio, USA

      @The Old Firm: I miss you.

    • The Old Firm profile image

      The Old Firm 8 years ago from Waikato/Bay Of Plenty, New Zealand

      nicompt. pick3 is to me also just a concept.

      The logics of simple mathematics will (sometimes) give me cause to pontificate.

      Logically, you shouldn't drink alcohol. I do.

      Logically, you shouldn't gamble . I do, as does any person who crosses the road, or buys a women a drink, or isn't catatonic, - or is.

      Spending the price of a Big Mac and chips on a lotto once a week and eating a home made sandwich instead seems to me a healthy investment. (Not buying anything and just eating salads is even better, look at the sex life of short lived rabbits!)

      Well, Gee Whoop,


    • sheila b. profile image

      sheila b. 8 years ago

      I've always figured I might as well throw my money away as to gamble it on the lottery. Some people buy their tickets and dream about winning, and I suppose that's a good fantasy, but how far do they fall when they don't win and have to face reality?

    • nicomp profile image

      nicomp really 8 years ago from Ohio, USA

      @breakfastpop: As they say "You can't win unless you play", which should be amended to be read "You can't win unless you play but even then you shouldn't expect to make your rent payment next month."

    • nicomp profile image

      nicomp really 8 years ago from Ohio, USA

      @drbj : political campaigns aren't much different than lotteries!

    • nicomp profile image

      nicomp really 8 years ago from Ohio, USA

      @The Old Firm: You're probably correct about the odds... given that I've never played Pick 3 myself... it makes sense that individual numbers can only be picked once per drawing.

    • breakfastpop profile image

      breakfastpop 8 years ago

      I hardly ever buy lottery tickets, but that doesn't stop me from wondering why I don't win every week!

    • drbj profile image

      drbj and sherry 8 years ago from south Florida

      nicomp - despite your intensive examination of lottery odds and persuasive argument against indulging in same, people will continue to bet the lottery because of their "somebody has to win - it might be me" rationalizing thinking.

      The same thinking, I might add, that leads folks to select leaders with loads of charisma and little leadership experience.

    • The Old Firm profile image

      The Old Firm 8 years ago from Waikato/Bay Of Plenty, New Zealand

      I consider buying a lottery ticket much like having a beer. Once the item is purchased (ticket/beer) your money's gone forever. With the beer your return is immediate, - a quenched thirst and if too enthusiastic, a hangover the next morning. With the ticket the return is the slight chance of winning something.(And if too enthusiastic probable poverty.)

      Combining the two and buying beer that has a promotion running of winning numbers under occasional caps is the best bet. You have a pathetic but valid reason to over-indulge and you just might win something for no extra outlay.

      BTW, with the "Pick 3" example aren't the odds 1:50*49*48 ?