How to Calculate Lottery Odds for 6/49 or Powerball

My recent Decoded Science article, “Winning Powerball Tickets in Arizona and Missouri: How to Calculate the (Slim) Odds“, takes aim at the wildly popular American powerball lottery.

How to Compute the Probability for  6/49 Compared to the American Powerball Lottery

"Taiwan Lottery Dream Sheet" by Prince Roy
"Taiwan Lottery Dream Sheet" by Prince Roy

The formula is actually the same for any lottery “without replacement and regardless of order“, and it’s already explained in my Decoded Science article.

The key difference comes from the number of balls in the urn, and how many are selected. To win the 6/49 jackpot requires selecting 6 numbers from the pool of 49. We need to calculate the number of combinations of 6 balls from 49:

  • The number of combinations = 49*48*47*46*45*44/(6!) = 49*48*47*46*45*44/(6*5*4*3*2*1) = 13,983,816.
  • Therefore the probability of winning the 6/49 jackpot is 1/13,983,816, by my calculations.

By comparison, the probability of winning the Powerball jackpot is 1 in 175,223,510. That’s about 12.5 times worse than the chance for winning 6/49.

Could We Compute the Odds of Winning Lotto Max or Other Canadian Lotteries?

Yes, of course we can compute the odds of winning any fair lottery “without replacement and regardless of order” using the same formula. We just need to learn the total number of balls in the pool, and how many are to be selected.

The Canada-wide Lotto Max jackpot requires selecting 7 numbers from a pool of 49, so the number of combinations is:

  • 49*48*47*46*45*44*43/(7!) = 49*48*47*46*45*44*43/(7*6*5*4*3*2*1) = 85,900,584 by my calculation.
  • Therefore Lotto Max has 85,900,584/13,983,816 = 6.14 times as many combinations as does the 6/49 lottery.

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Thank you for reading about the probability of winning Lotto 4/49 or the American Powerball lottery.

Disclaimer: The information contained in this article is for educational purposes only and should not be used to encourage gambling or gaming activities; certainly not to the point where it interferes with normal household finances.

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