How to Calculate Lotto Odds

Step-by-Step Guide

1. 1

   Step 1: Understand the calculations involved.

   The odds of winning any lottery where numbers are chosen from a set, provided order doesn't matter, are defined by the formula n!r!(n−r)!{\displaystyle {\frac {n!}{r!(n-r)!}}}.
   
   In the formula, n stands for the total number of possible numbers and r stands for the number of numbers chosen.
   
   The "!" denotes a factorial, which for any integer n is n*(n-1)*(n-2)...and so on until 0 is reached.
   
   For example, 3! represents 3×2×1{\displaystyle 3\times 2\times 1}.
   
   For a simple example, imagine you have to choose two numbers and you can pick numbers from 1 to
   5.
   
   Your odds of choosing the two "correct" numbers (the winning numbers) would be defined as 5!2!×3!{\displaystyle {\frac {5!}{2!\times 3!}}}.
   
   This would then be solved as 5×4×3×2×12×1×3×2×1{\displaystyle {\frac {5\times 4\times 3\times 2\times 1}{2\times 1\times 3\times 2\times 1}}}, which is 120÷12{\displaystyle 120\div 12}, or
   10.
   
   So, your odds of winning this game are 1 in
   10.Factorial calculations can get unwieldy, especially with large numbers.
   
   Most calculators have a factorial function to ease your calculations.
   
   Alternately, you can type the factorial into Google (as "55!" for example) and it will solve it for you.
2. 2

   Step 2: Establish the lottery's rules.

   The majority of "money millions," Powerball, and other large lotteries use roughly the same rules. 5 or 6 numbers are chosen from a large pool of numbers in no particular order.
   
   Numbers may not be repeated.
   
   In some games, a final number is added on the end (the Powerball in Powerball games is an example).
   
   Using the standard Powerball rules, we see that 5 numbers (not including the Powerball) are chosen from 69 possible numbers.Other games may have you choose 5 or 6 numbers, or more, from a larger or smaller pool of numbers., The first part of the Powerball odds are calculated as the odds of correctly choosing the first five numbers.
   
   This is described handily by the probability formula introduced earlier.
   
   So, for these specific rules, the completed equation would be: 69!5!(69−5)!{\displaystyle {\frac {69!}{5!(69-5)!}}}, which simplifies to 69!5!×64!{\displaystyle {\frac {69!}{5!\times 64!}}}., Solving this equation is best done entirely in a search engine or calculator, as the numbers involved are inconvenient to write down between steps.
   
   When solved, the equation should give 11,238,513.
   
   This means that you have a 1 in 11,238,513 chance of choosing the five numbers correctly., Now, to include the odds you'll choose the Powerball correctly and win the jackpot, you'll simply have to multiple the number from your previous result by the size of the Powerball number pool.
   
   For the standard game, there are 26 possible Powerball numbers.
   
   So, multiply your previous result, 11,238,513 in this case, by the final number, which is 26, to get your final odds, which are 292,201,338.
   
   So, your odds of choosing the first five numbers and the Powerball correctly are 1 in 292,201,338.
3. 3

   Step 3: Input the numbers into the probability equation.

4. 4

   Step 4: Calculate your odds of choosing correctly.

5. 5

   Step 5: Multiply to include a final number.

Detailed Guide

About the Author