Original Blog Entry: Published odds and the permutation factor

hypersoniq — Wed, 14 May 2025 15:06:37 GMT

On a simple game like the pick 3, odds are super easy to calculate... it is 10, representing each possible digit, multiplied by each other...

Such that 10x10x10 =1,000 possible combinations from 000 to 999

This holds for pick 4

10x10x10x10 = 10,000 possible combinations from 0000 to 9999

And pick 5 as well

10x10x10x10x10 = 100,000 possible combinations from 00000 to 99999

When we apply the same to the top prize on the powerball, things change a bit. In a replacement draw, any number can only be selected once in the first 5... so the raw odds calculation for the 5/69 + 1/26 looks like

69x68x67x66x65x26 = 35,064,160,560

That is over 35 Billion possible combos, but their odds are published as 1 in 292,201,338

Why?

Permutations!

Taking the raw odds calculated first, and dividing by the published odds gives you a whole number of 120.

Therefore we must consider that there are 120 ways to display a pick other than sorted order.

Let's look at another example... the cash 4 life (MUSL multi state game, called cash 4 life in PA)

We will look at the second prize (1,000 a week for life), paid out for matching 5/60)

Raw odds

60x59x58x57x56 = 655,381,440

Published odds = 1 in 7,282,016

Permutation factor (raw/published) is 90.

The odd part is that they do not always result in a whole number for a permutation factor.

Looking at the powerball second prize (pick 5 white balls from 69)

Raw odds

69x68x67x66x65 = 1,348,621,560

Published odds = 11,688,054

Which when dividing raw vs published gives

115.384610646

Which may mean there are more factors than permutation when calculating published odds...

I wonder why?... [ More ]