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The winning Powerball tickets were sold in Arizona and Missouri – if you didn’t win, you may be kicking yourself for wasting money on a ticket, but the odds of winning are always better if you buy in.

What are those odds, anyway, and how do we calculate them?

## Calculating the Powerball Odds

It’s easy to learn the poor odds of becoming incredibly wealthy by winning the Powerball lottery, but just how do they calculate the odds of winning the ultimate get-rich-quick scheme?

The current rules for a Powerball lotto’s grand prize win are fairly simple. The winning ticket must predict the values of five white balls chosen *without replacement* from a group of 59, and also predict the value of one red ball chosen from a group of 35.

The official Powerball site gives the probability of winning the $2 play as “*1 in 175,223,510*” to match all 5 white plus 1 red ball.

They don’t show how they calculate the number of possible draw combinations:

- There are ( (59*58*57*56*55)*35 )/ (5!) = ( (59*58*57*56*55)*35 )/ (5*4*3*2*1) combinations.
- According to my computer’s calculator, this equation equals 175,223,510.

The probability of winning is therefore one out of all the possible combinations, or 1/175,223,510.

What does this calculation of combinations mean?

## The Formula for the Probability of a Lucky Draw in a Lottery

The general formula for the number of combinations when drawing ‘*m*‘ correct items from a set of ‘*n*‘, without replacement and regardless of the order, is:

C = n*(n – 1)*(n – 2)*…(n – m – 1)/(m!), where…

- ‘C’ is the number of combinations;
- ‘n’ is the number of items in the set;
- ‘m’ is the number of items to be selected.

So we have ‘n’ choices for the first item, but only (n-1) choices for the second selection. Why? We’ve already taken away one item.

When we take away the *m*-th item, there are only (m – 1) items remaining for the selection. Thus the second factor is (m-1). This continues until all ‘m’ factors have been multiplied together.

Then we divide by “m!”, or “*m* factorial”, which was discussed in “Introducing the Factorial: the Exclamation Mark of Math“.

The five white balls in Powerball count down from 59 through 55 and the “5!” divisor in the main calculation. The number 35 accounts for the solitary red ball.

Mike DeHaan applies his Bachelor of Math in Computer Sciences degree, years of Cobol programming and quality assurance (including testing credit card interest calculations) to research and present mathematical theory for the layman.

Here in Decoded Science, Mike’s main focus is mathematics. He covers basic mathematical theory, uncovers paradoxes, applies calculations to popular movies, and reports on math news.

Mike began writing professionally in 2010 as the sole proprietor of DeHaan Services.

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