Calculate Lotto Odds

Everyone's heard comparisons between the odds of winning the lottery and the odds of other unlikely events, like getting struck by lightning. It's true, the odds of winning the jackpot on a game like Powerball or another pick-6 lottery game are incredibly low. But just how low are they? And how many times would you have to play to have a better chance of winning? These answers can be found down to the exact odds with some simple calculations.

Steps

Calculating Jackpot Odds

  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 <math>\frac{n!}{r!(n-r)!}</math>. 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 <math>3\times2\times1</math>.
    • 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 <math>\frac{5!}{2!\times3!}</math>.
    • This would then be solved as <math>\frac{5\times4\times3\times2\times1}{2\times1\times3\times2\times1}</math>, which is <math>120\div12</math>, or 10.
    • So, your odds of winning this game are 1 in 10.[1]
    • 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. 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.[2]
    • Other games may have you choose 5 or 6 numbers, or more, from a larger or smaller pool of numbers.[3]
  3. Input the numbers into the probability equation. 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: <math>\frac{69!}{5!(69-5)!}</math>, which simplifies to <math>\frac{69!}{5!\times64!}</math>.[2]
  4. 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. 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.[2]
  5. Multiply to include a final number. 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.[4]

Determining Lesser Prize Odds

  1. Start with the jackpot odds. In most cases, there are lesser prizes available for choosing some of the numbers correctly. Choosing 3 or 4 of the winning numbers might yield you a payout of several hundred or thousand dollars. In this case, your odds are calculation as the odds of choosing a few right numbers and your odds of choosing an opposing amount of wrong numbers combined. This requires knowing the total number of possible combinations first. These can be described with the odds of choosing all of the winning numbers correctly.[1]
    • You'll need to have first solved for the jackpot odds as described in the method "Calculating Jackpot Odds."
    • To simplify our calculations, we will just use the odds of correctly choosing the first five Powerball numbers. In the other method, this was found to be 1 in 11,238,513.
  2. Set up your equation. Your odds of winning "k" of the numbers "r" from the total pool of numbers "n" can be defined as: <math>\frac{\frac{r!}{k!(r-k)!}\times\frac{(n-r)!}{(r-k)!(n-r-k)!}}{\frac{n!}{r!(n-r)!}}</math>. This formula seems much more complicated, but is really just three copies of the simple probability equation described in the other method. Simply input your numbers for n, r, and k.
    • For example, your odds of winning 3 of 5 numbers from a pool of 69 would be described by the equation <math>\frac{\frac{5!}{3!\times2!}\times\frac{64!}{2!\times61!}}{\frac{69!}{5!\times64!}}</math>.
  3. Solve for the odds of winning. Just as with the base equation, this equation is best solved by typing the entire thing into a calculator or search engine, as some intermediate numbers involved in the calculation are too large to write down. If solved correctly, the result will be the odds that three of the five Powerball numbers are chosen correctly.
    • For the example, this would be 579.76. So, your odds of choosing 3 of the 5 correctly are 1 in 579.76.
  4. Change your desired number of winning numbers. You can edit your calculation by changing the value of k to find the odds of winning different portions of the five numbers. Your odds of winning will decrease as k increases and vice-versa.[1]

Calculating Other Lotto Odds

  1. Find the expected value of a lottery ticket. The expected value of a lottery ticket represents the theoretical return from purchasing one lottery ticket. In other words, it is what you can theoretically expect to get back from buying one ticket. It can be calculated by multiplying the odds of a certain payout (jackpot, 4 numbers correct, 3 numbers correct, etc.) by the value of the payout and then adding them up. This number is obviously skewed by the massive jackpot, however.
    • Most of the time, your payout will be much less than the expected value.
    • For the standard choose 5+1 of 69 and 26 Powerball, the expected value of a ticket is about $1.78.[5]
  2. Compare cost to expected value. You can determine the expected benefit of playing the lottery by comparing the expected value of a ticket to the cost of a ticket. Most of the time, it will be much less. In addition, the actual return expected differs greatly from the expected value, despite the name. Most people will only get a fraction of the ideal expected value, if anything at all.
    • However, some lottery games have better payouts than others.
    • For example, at one time, the New York lottery's $1 Take Five ticket had an expected value equal to its cost. This meant that by playing this lottery, players could expect to break even.[6]
  3. Determine the increase in odds from playing multiple times. Playing the lottery multiple times can increase your overall odds of winning, however slightly. It's easier to envision this increase as a decrease in your chance of losing. For example, if your overall chances of winning are 1 in 250,000,000, your chances of losing on one play are <math>249,999,999\div250,000,000</math>, which is equal to a number very close to 1 (0.99999...). If you play twice, that number is squared (<math>(249,999,999\div250,000,000)^{2}</math>), representing a movement slightly away from 1 (and a better chance of winning).[4]
  4. Find the number of plays needed for goods 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, the increase is not significant for a long time. For example, using the odds above (1 in 250 million on one play), it would take roughly 180 million plays to reach 50-50 odds of winning.[4]
    • At this rate, if you bought ten tickets a day for 49,300 years, you would have a 50 percent chance of winning.
    • This does not mean, however, that buying two tickets on that day would guarantee a win. Instead, your overall odds of winning would remain at roughly 50 percent.



Tips

  • Any set of numbers has exactly the same odds as any other set. 32-45-22-19-09-11 is no different than 1-2-3-4-5-6.
  • Don't fall for lottery scams where somebody tells you they have a sure-fire way of winning. If someone had a guaranteed, sure-fire way of winning, it would be self-defeating to tell you about it.

Warnings

  • If you think you have a problem with gambling, you probably do. Gamblers Anonymous is a good source of information and help for those afflicted with gambling addiction.
  • Don't gamble more than you can afford to lose.

Related Articles

Sources and Citations

  1. ↑ 1.0 1.1 1.2 http://garsia.math.yorku.ca/~zabrocki/math5020f03/lot649/lot649v3.pdf
  2. ↑ 2.0 2.1 2.2 http://www.durangobill.com/PowerballOdds.html
  3. ↑ http://www.flalottery.com/exptkt/lottoodds.pdf
  4. ↑ 4.0 4.1 4.2 http://www.quickanddirtytips.com/education/math/what-are-the-odds-of-winning-the-lottery
  5. ↑ http://www.businessinsider.com/powerball-lottery-jackpot-statistics-probability-2013-5
  6. ↑ http://wmbriggs.com/post/5285/
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