Figure Your Odds of Holding a Winning Lottery Ticket

If you buy one random drawing lottery ticket, you have a chance of winning. What are your chances of winning though? If you buy more than one lottery ticket, how much does this improve your odds of having at least one winning ticket? In this article, you'll learn how to answer these questions and add a little bit of clarity to what is, to most people, a shot in the dark. The first method below is the most accurate, but it takes some time. The second method presented is simpler and faster, and will approximate the odds closely enough for most people--in fact the answer is the same for the example below.

Steps

  1. Gather the following information:
    • T = How many lottery tickets will be sold
    • W = How many tickets sold will be winners
    • P = How many tickets you are planning to buy
  2. Write down your first fraction: (T-W)/T. For the moment, don't simplify the fraction by reducing it to lowest terms.
    • For example, suppose the lottery will sell 23,000,000 tickets, there will be 1,000 winning tickets, and you are planning to buy 6 tickets. Then T = 23000000, W = 1000, and P = 6. The first fraction will be (23000000-1000)/23000000 = 22999000/23000000.
  3. Simplify the fraction (punch 22999000 divided by 23000000 into a calculator to get a decimal, in this case 0.99995652173913043478260869565217) and jump to the Subtract step if you only bought one ticket. If you bought more than one ticket, skip this step.
  4. Write more fractions by reducing the previous numerators and denominators by one, until the total number of fractions is equal to P, the number of tickets you are planning to buy. If you bought 6 tickets in the above example, these are the fractions you should come up with:
    1. 22999000/23000000
      • This is your first fraction, from an earlier step.
    2. 22998999/22999999
      • This is the result of subtracting one from each of the numbers in your first fraction; the following fractions continue this pattern.
    3. 22998998/22999998
    4. 22998997/22999997
    5. 22998996/22999996
    6. 22998995/22999995
  5. Multiply all of the fractions together. There are two ways to do this. You can multiply all of the numerators, then multiply all the denominators, and then divide the numerator product by the denominator product. Or, if you have a calculator that can generate long decimals (such as the standard calculator in your computer), simplify all of the fractions (numerator divided by denominator to generate a decimal) and then multiply all of the decimals together. The resulting number is the probability of having NO winners among the tickets you purchased.
    • The 5 fractions in this example generate these decimals, respectively:
      • 0.99995652173913043478260869565217
      • 0.99995652173724007553217719705118
      • 0.99995652173534971611736661890145
      • 0.99995652173345935653817693976221
      • 0.99995652173156899679460813819272
      • 0.99995652172967863688666019275222
    • Since you have 5 fractions, you must multiply the resulting decimals together, which in this case results in 0.99973915876017716698091198324496
  6. Subtract the result from 1 to get the probability of having AT LEAST ONE winner among the tickets you purchased.
    • 1 - 0.99973915876017716698091198324496 = 0.000260841239822833019088016756
    • For ease of calculation, let's cut it down to .0002608412
  7. Invert the fraction. Press the "1/x" or "x -1" button on your calculator to invert the fraction you calculated in the previous step. 1 / .0002608412 ~= 3834, meaning that you have about a 1 in 3834 chance of holding at least one winning ticket. If your calculator doesn't have this function, skip this step and move on to the next.
  8. Convert the decimal into a fraction. Count the number of characters after the decimal point, as that will be the number of zeros following 1 in your denominator; to get the numerator, take the decimal point and any preceding zeros away.
    • .0002608412 has 10 characters after the decimal point, so your denominator is 10,000,000,000 (1 followed by 10 zeros)
    • Without the decimal and the preceding zeros, .0002608412 = 2608412
    • Your fraction is 2,608,412/10,000,000,000
  9. Calculate your chances. Divide the denominator by the numerator. In this case, 10,000,000,000 divided by 2,608,412 equals 3,834 (when rounded). This corresponds to about 1 chance in 3,834 of holding at least one winning ticket.

Simplified Method

  1. Use this equation: Chance of Win = [1 - (1 - W/T)^P] * 100, which for P = 1 would reduce to: Chance of Win = W/T * 100. The resulting number is your percentage of winning. Using the numbers from the example above, this equation tells us that we have a 0.026084% (less than three one-hundredths of a percent) chance of winning.
  2. Convert to a fraction as above. Before multiplying by 100 to obtain the percentage, you will have calculated ~0.00026084. Create a fraction as from this as in the last steps of the first method above, so that you have 26084/100,000,000 (8 zeros after the one in this case because there are only 8 digits after the decimal point). Divide the denominator by the numerator, and your answer, when rounded, is 3,834--the same number you would have arrived at by use of the longer method.

Tips

  • Some lotteries allow multiple wins if more than one ticket is purchased. These methods simply calculate your chance of at least one win.
  • Some lotteries provide the odds of winning, so you can check your math.
  • If the odds of winning with any given ticket are low, and you are only buying a few tickets, you can get a good estimate of your odds by multiplying the number of tickets you purchase by the odds of winning for each one (the number of winners divided by the number of tickets to be sold). If, for example, you buy 6 tickets in a 23,000,000 ticket lottery with 1,000 winners, then the odds of any one ticket being a winner is 1,000/23,000,000 or 1/23,000, so the probability of you holding a winner is 6/23,000 (about 1 in 3833, which is still about 0.00026 or 0.026%). This estimate slightly overstates your odds of winning, but if your chances are low enough, the error will be very small.
  • These calculations are only valid for a random drawing lottery. A lottery based on matching numbers would have an entirely different calculation, based on the number of balls drawn and the number of total balls to draw from.

Warnings

  • If playing a lottery is anything more than just fun for you, and you are thinking of risking money you cannot afford to lose, you are advised to avoid playing a lottery.

Things You'll Need

  • A calculator

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