All the chances of winning on the UK National Lottery

The sixth Murderous Maths book Do You Feel Lucky? explains the chances of winning on the UK National Lottery. When the book was first published in 2001, players would pick 6 numbers from 1-49. The chances of all six numbers being drawn to win the Jackpot were 1 in 13,983,816. You could also win prizes if you match three, four, five or "five plus the bonus ball".

Since October 2015, the game has changed. Players now pick six numbers from 1-59. Just by raising the numbers available from 49 to 59, it made the jackpot more than three times harder to win!

Here are the approximate chances of matching the drawn numbers: JACKPOT (matching all six numbers): 1 in 45 million FIVE + THE BONUS BALL: 1 in 7.5 million FIVE: 1 in 150,000 FOUR: 1 in 2,000 THREE: 1 in 100 TWO: 1 in 10 And just for interest, here are some other chances: The chance of winning ANY prize*: 1 in 9.2 (nearly 11%) The chance of matching just ONE number: 1 in 2.5 (nearly 40%) The chance of not matching ANY numbers: 1 in 2 (about 51%) * You need to match two or more numbers to win a prize.

If you buy one lottery ticket every minute for the next 90 years, the chances are that ONE ticket will win the lottery jackpot.

How to calculate the different chances

WARNING..... If you haven't read "Do you feel lucky?" you might find some of the sums on this page a bit scary!

The main thing to know is that the total number of ways of choosing your six numbers is:

Before you read on, would you like to know where the number 45,057,474 comes from?

"It's about the same as picking a single golden grain from a huge bath full of rice" - Sir David Spiegelhalter, World Loop Champion and Winton Professor for the Public Understanding of Risk, University of Cambridge

JACKPOT CHANCE: only ONE of these ways will win you the jackpot, so
the jackpot chance is 1 in 45,057,474 or about 1 in 45 million

MATCHING THREE NUMBERS: we have to work out how many ways you can select your six numbers so that three of them match up and three of them don't.

To start with, of the six numbers that are actually drawn, you need a combination of three of them, so how many combinations of 3 can you choose from 6? It's:
Your other three numbers must come from the 53 numbers that are not drawn, so the number of combinations of 3 you can choose from 53 is...

A big HURRAH for Simon Ferguson who corrected the numbers in this section for us!

To get the total number of combinations that include 3 matching numbers and 3 that don't match, you multiply these answers together.

You get 20 x 23426 = 468,520. That's how many different ways you can match three winning numbers, so your chances of matching 3 numbers are 468,520 out of the 45,057,474 possible ways the numbers could come up. Obviously this looks a bit mind boggling, so to make it simpler you divide 45,057,474 by 468,520 to work out that the chance of matching 3 numbers is 1 in 96.167 which is about 1 in 96.

MATCHING FOUR NUMBERS: this time we need to see how many ways there are of picking four numbers from the winning six, and then multiply this by how many ways there are of picking two numbers from the 53 losers.

The sums look like this:

As there are 20,670 ways of matching four winning numbers, your chances are 20,670 in 45,057,474. It works out that the chance of matching 4 numbers is 1 in 2,180

MATCHING FIVE NUMBERS: this time we need to see how many ways there are of picking five numbers from the winning six, and then multiply this by how many ways there are of picking one number from the 53 losers. (This last bit's easy: if you have a choice of 53 things and you can choose one, how many choices have you got? 53 of course!)

Here comes the sums:

This means there are 318 ways of matching five winning numbers, but before we look at the chances, remember that with some of these ways your sixth number will also match the bonus number.

MATCHING FIVE NUMBERS AND THE BONUS: The sums here are slightly different. Have a think about this:

You've just seen the main six numbers drawn on the lottery. You've matched five of them, and are leaping about the room and kissing the telly. The trouble is that ONE of your chosen numbers did not match up.

We've just seen that there are 318 combinations of five numbers that will match. The seventh "Bonus" number is about to be drawn, so what are the chances of your unmatched number matching the bonus? As 6 numbers have already been drawn from the 59, there are there are 53 numbers left that the bonus can be chosen from, so it's 1 chance in 53.

We've already worked out that there are 318 combinations that match five balls, and 1 in 53 of these combinations will have the sixth ball match the bonus. Therefore the number of combinations that match five numbers PLUS the bonus is 1/53 x 318 = 6.

This leaves 318 -6 = 312 chances that match five numbers but NOT the bonus.

So for just five numbers, the chances are 312 in 45,057,474 which means the chance of matching 5 numbers without the bonus is 1 in 144,415

And for five numbers and the bonus, the chances are 6 in 45,057,474 which gives us the chance of matching 5 numbers plus the bonus is 1 in 7,509,579

Where does 45,057,474 come from?

The way the UK lottery works is that you have a card with the numbers 1-59 marked on it and you have to choose any six of them. What we need to work out is how many different ways you can choose 6 numbers out of the 59 available.

In maths we say "how many different combinations of 6 can we choose from a total of 59" and we write it with the numbers next to a big letter "C" as you can see in the diagram. To work it out you fill the numbers into the formula. (You can find out exactly where this groovy looking formula comes from in Do You Feel Lucky?)

To use the formula you need to know that the "!" sign is called a "factorial" which means you need to multiply the number by every smaller number down to 1. So for example 5! = 5x4x3x2x1 = 120. (Some calculators have a ! button on them.) If we wrote the top line of this formula out in full we'd get 59x58x57x56.... right down to ...4x3x2x1. The bottom line is 6x5x4x3x2x1 x 53x52x51x50...etc down to ...4x3x2x1.

It all seems awful but DON'T PANIC! The fun part is that monstrous sums like these cancel themselves to bits and you can actually rip though them really quickly. This one produces the answer that there are 45,057,474 combinations in total, and only ONE of them will win you the jackpot. So your chances of winning the jackpot are 1 in 45,057,474.

Now you've understood all this, let's get back to the top