The mathematical truth behind winning Lotto


Seriously, what are the chances of hitting the jackpot?

Most people think that mathematics is hard and that you need to go to university to understand some of the stranger concepts. Actually, higher order maths is quite simple once you break down the principals involved into simple tricks. In this article I will deconstruct one of the basic concepts in probability, one that people may not realise is used often– factorials and how to calculate the odds of winning a jackpot. I will also explain how out of all the lotteries, Powerball is the worst to play.

Factorials are a repeating sequence of multiple numbers. As a principal in probability, factorials represent the chance of a specific sequence of numbers being drawn in a lottery. For example, let us suppose we have a pool of 10 numbers and we want to draw out from that pool the numbers 6, 2 and 5 in that order. The first number (6) has a 1/10 chance of being drawn from the pool first, the odds of this is 1/10. Once 6 is drawn the second number (2) has a 1/9 chance of being drawn, the odds of this occurring is 1/(10 x 9) = 1/90.

After 2 is drawn the third number (5) has a 1/8 chance of being drawn, the total odds of this arrangement is 1/(10 x 9 x 8), this equals 1/720. The denominator of the fraction demonstrates how a factorial works, a sequence of consecutive numbers that multiply. If you apply this to the Saturday Lotto for example, the odds are 1/8,145,060.

My personal favourite example of probability and lottery is, in fact, a scene from the cult science-fiction show Red Dwarf in which one of the characters, Dave Lister, is injected with a luck virus. To prove he is infected, he shuffles a deck of cards and produces every ace card. His team member, the android Kryten, comments the chances of doing so every time: “13 to one! 221 to one! 5,525 to one! 270,725 to one! Well done, Sir!”

But why is Powerball the worst to play?

Powerball has been in Australia now for over 20 years and the game has consistently made news for offering the biggest jackpots. That is not a coincidence – it is the result of the powerball, the additional ball at the end of a draw. If you are not familiar with this lottery draw, Powerball’s structure is unique in that not only does the draw select in this case seven balls from one barrel but it also selects one extra ball, the powerball as it were, from a second independent barrel. This extra ball blows out our calculations and makes it in my opinion, the most difficult lottery game ever.

Originally drawing five balls from 45 and one powerball from 45, the odds of winning a Powerball lottery draw were 1/54,979,155, exactly 6.75 times lower than the Saturday Lotto example which has not changed at all. In 2013 Tatts Group decided to change the conditions of the draw by decreasing the number of balls in both barrels and increase the number of drawn balls. This changed the variables and the probability shrank to 1/76,767,600.

The basic reason for this is the extra ball increases the length of the factorials in the formula and therefore multiplies larger numbers. This fuels the decrease in the odds while at the same time increasing the chance of winning a minor prize. Therefore, when in 2018 Tatts Group again changed the structure of the draw the odds decreased again; at the time of writing this article, the probability of winning a Powerball jackpot is 1/134,490,400 – nearly half as likely two years ago and roughly 41% of the chance seven years ago.

While this piece is not written to discourage you from playing the lotto, I feel it is reasonable to understand not only what the chances of winning are, but also how you personally can discover this yourself. I quite like playing the lotto now and then, but my personal advice is stick to Saturday Lotto. Also play the Mersenne primes, but that is for another time.