Understand the calculations involved. To find the odds of winning any lottery, divide the number of winning lottery numbers by the total number of possible lottery numbers. If the numbers are chosen from a set and the order of the numbers doesn't matter, use the formula . 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 .
Establish the lottery's rules. The majority of Mega 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 chosen from a smaller set of numbers (the "Powerball" in Powerball games is an example). In Powerball, 5 numbers are chosen from 69 possible numbers. Then, for the single Powerball, one number is chosen from a set of 26 possible numbers.
Input the numbers into the probability equation. The first part of Powerball odds determines the number of ways 5 numbers could be chosen out of 69 unique numbers. Using Powerball rules, the completed equation for the first 5 numbers would be: , which simplifies to .
Calculate your odds of choosing correctly. Solving this equation is best done entirely in a search engine or calculator, as the numbers involved are inconvenient to write down between steps. The result tells you there are 11,238,513 possible combinations of 5 numbers in a set of 69 unique numbers. This means that you have a 1 in 11,238,513 chance of choosing the five numbers correctly.
Multiply to calculate your odds of winning the jackpot. To calculate the odds that you'll guess the first 5 numbers and the Powerball correctly to win the jackpot, multiply the odds that you'll guess the first 5 numbers (1 in 11,238,513) by the odds that you'll guess the Powerball correctly (1 in 26). Your equation would be .
Calculate your odds of winning the second prize. To return to the Powerball game, you have 5 numbers and a single Powerball. If you guess all 5 of the other numbers correctly but don't get the Powerball, you'll win the second prize. If you calculated your odds of winning the jackpot, you already know that your odds of guessing all 5 numbers correctly are 1 in 11,238,513.
Use an expanded equation to find your odds for other prizes. To win other prizes, you guess some, but not all, of the winning numbers correctly. To figure out your odds, use an equation in which "k" represents the numbers you choose correctly, "r" represents the total numbers drawn, and "n" represents the number of unique numbers the numbers will be drawn from. Without numbers, the formula looks like this: .
Solve your equation to find the odds of correctly guessing the numbers. Just as with the base equation, this equation is best solved by typing the entire thing into a calculator or search engine. Some intermediate numbers involved in the calculation would be cumbersome to write down and it would be easy to make a mistake.
Multiply the result by the Powerball value to determine your odds of winning that prize. While this formula gives you the odds of guessing only some of the numbers correctly, you still haven't factored in the Powerball. To find your true odds, multiply the result by your odds of getting the Powerball number correct or incorrect (whichever value you want to find).
Change the number of correctly guessed numbers for other prizes. Once you have the formula down, simply change the value of "k" to find the odds of winning different levels of prizes. Generally, your odds of winning will decrease as the value of "k" increases.
Find the expected return of a lottery ticket. The expected return tells you what you could theoretically expect to get back in return for buying a single lottery ticket. To calculate the expected return of a single ticket, multiply the odds of a particular payout by the value of that payout. If you did this with every possible prize you could win, you would get a range of expected returns.
Compare the cost of a single ticket to its expected return. You can determine the expected benefit of playing the lottery by comparing the expected return of a ticket to the cost of a ticket. Most of the time, the expected return will be lower than the cost of the ticket. Additionally, your actual return will likely differ greatly from the expected value. You'll typically only get a fraction of the expected value if anything at all.
Determine the increase in odds from playing multiple times. Playing the lottery multiple times increases your overall odds of winning, however slightly. It's easier to envision this increase as a decrease in your chance of losing.
Find the number of plays needed for decent odds of winning. Most lottery players are convinced that if they play often enough, they will significantly increase their chances of winning. It is true that playing more increases your odds of winning. However, it takes a long time for that increased chance to become significant.