Winning Powerball Tickets in Arizona and Missouri: How to Calculate the (Slim) Odds


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Winning the Powerball lottery isn’t simple.  Image by Mike DeHaan

A Simple Example of the Probability Calculation for a Lottery

A smaller and simpler example may help.

Let’s examine the combinations when selecting three balls from six, numbered 1 through 6.

As shown in the image, there are four combinations that start with “1, 2”; three that start with “1,3”; and so on.

The grand total is 20 combinations.

6*5*4/(3!) = 6*5*4/6 = 20.

Again, the probability of any one of these combiniations being chosen by a fair and random selection is “one divided by the number of combinations,” or 1/20.

Playing The Powerball Lottery: Is it Worth It?

The powerball lottery odds only change if the number of balls changes, or some other rule is amended. An economist might calculate the utility of buying a $2 play in Powerball as the difference between the cost and the expected value. The “expected value” equals the probability of winning, times the maximum payout amount. Therefore, if the grand prize is over $350,447,020, then an economist would argue that it is rational to gamble $2.

Powerball retailers and advertisers would be quick to add that a person with no tickets has absolutely no chance of hitting the jackpot. Spending two dollars (that we assume are not needed for living expenses) raises the odds of winning Powerball significantly. Going from zero to “1 in 175,223,510” with a minimal investment obviously made sense for the owners of the winning Powerball tickets, sold in Arizona and Missouri – they’ll be splitting the winnings with one another and the IRS, but the payout is still significant.


Mungin, L. Winning Powerball Tickets Sold in Missouri, Arizona. (2012). CNN. Accessed November 29, 2012.

Official Powerball web site. (How to Play) Powerball: It’s America’s Game and Powerball – Prizes and Odds. (2012). Accessed November 29, 2012.

Turpin, Zachary. The Secret Behind Powerball Strategies. Book of Odds. Accessed November 29, 2012.

Note: Turpin claims there are 39 red balls, rather than 35. That increases the number of combinations to  195,249,054. This article uses the numbers given today by the official site; but the author began with Turpin’s formula for calculating the odds of winning powerball, verified the formula, then inserted the official numbers to calculate the powerball odds.

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