Lottery Results

Lotto 6/49 Winning Numbers

Unfortunately, the simple conclusion regarding Lottery 6/49 winning odds is that they are vanishingly small.

The chances of winning a lottery jackpot are principally determined by several factors: the count of possible lottery numbers, the count of winning lottery numbers drawn, whether or not order is significant and whether the drawn numbers are returned to the 'bag' or not.

In a typical 6/49 lotto, 6 (k) numbers are drawn from a range of 49 (n) and if the 6 numbers on your lottery ticket match the numbers drawn, you are a jackpot winner - this is true no matter the order in which the numbers appear. The odds of this happening by the way are 1 in 14 million (13,983,816 to be exact). So, why are the chances of winning the lottery so slim?

Let's work through an example. If you start with a bag of 49 differently-numbered lottery balls, clearly you have a 1 in 49 chance of predicting the number of the 1st ball out of the bag. Looking at it in a different light, there are 49 different ways of choosing that first lottery number. When you come to draw the 2nd number, there are now only 48 balls left in the bag (in case of no return of the drawn balls to the bag), so you have a 1 in 48 chance of predicting this number (i.e. there are 48 different ways of choosing that second number).

Thus, each of the 49 ways of choosing the first number has 48 different ways of choosing the second. This means that the odds of correctly predicting 2 numbers drawn from 49 is calculated as: 49 x 48. On drawing the third lottery number you only have 47 ways of choosing the number; but of course you could have gotten to this point in any of 49 x 48 ways, so the chances of correctly predicting 3 numbers drawn from 49 is calculated as: 49 x 48 x 47. And so it goes on until the sixth number has been drawn, giving the final calculation: 49 x 48 x 47 x 46 x 45 x 44 (also written as 49! /(49-6)!). This works out at a really scary number (10,068,347,520) but clearly a whole lot bigger than the 14 million we were talking about above. So how do we get to that final figure of 1 in 13, 983,816?

The last step we need to take is to understand that the order of our 6 numbers is not significant. That is, if your ticket says 01 02 03 04 05 06, then you'll be popping the champagne so long as all the numbers 1 through 6 are drawn, no matter what order they come out. To put it another way, given any set of 6 numbers, there are 6 x 5 x 4 x 3 x 2 x 1 = 6! = 720 ways they could be drawn. Dividing 10,068,347,520 by 720 gives 13,983,816, also written as 49! / (6!·(49-6)!), or more generally as

To put this number in context, let's say that you're immortal and are going to play 1 ticket with the same numbers every week forever. Since 13,983,816 weeks is roughly 269,000 years you probably would win the jackpot only once in your first quarter-million years! To add insult to injury, your heroic patience and faithful play over the millennia will be cruelly rewarded, for you probably will have spent two to three times more money buying tickets than you will receive at long last with your one jackpot win.

Alternatively, imagine a computer randomly drawing 6 lottery numbers every second of every day. Starting it off at 1 second past midnight on January 1st, you would have to wait until 8:23pm on June 11th before it had executed 13,983,816 times. That is, picking the 6 winning numbers is as hard as picking a single second out of more than 5 months!