How to work out all the chances of winning on the
UK National Lottery

Euromillions, US Powerball and other Lottery Chances

THE GAME:

You select six numbers from 1-59. Six numbers are drawn plus a seventh "bonus ball".

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
86 years,
the chances are that
ONE
ticket will win the UK lottery jackpot.


What's more likely than winning the jackpot?


How to calculate the different chances

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 = 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...


What's more likely than winning the jackpot?
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 = 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 = 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:

What's more likely than winning the jackpot?
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 = 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 = 1 in 7,509,579


What's more likely than winning the UK National Lottery Jackpot?


You're 75 times more likely to get a royal flush on your first ever hand of poker than win the UK National Lottery jackpot.

The odds of winning the UK National Lottery are 1/45,057,474. If you just buy one lottery ticket in your life, these things are more likely to happen to you...
  • Winning an Olympic gold medal is almost 70 times more likely (chance= 1/700,000)
  • Being hit by something falling from the sky is almost five times more likely (chance=1/10,000,000)
  • Being hit by lightning is almost five times more likely (chance=1/10,000,000)
  • Becoming an astronaut is about four times more likely (chance= 1/12,000,000)
  • Playing your first hand of poker and getting a royal flush is 75 times more likely (chance= 1/650,000)

One FINAL thought about the lottery:

Assuming your life expectancy is 75 years, on average you should hope to be alive for a total of about 40,000,000 minutes. This means that your chances of dying in any particular minute are 1/40,000,000. Therefore, if you want a lottery ticket, be sure to buy it in the last minute before the draw is made. If you leave a gap of longer than a minute, then you are more likely to die while you are waiting for the draw than you are to win the jackpot.

Good luck!


The chances of other National Lottery Jackpots

When comparing the chances of winning on different lotteries, you should also factor in the returns. As a simple example, if you have a 1 in 10 chance of winning $100 this isn't as good as a 1 in 100 chance of winning $5,000. Also, different lotteries cost different amounts to play, jackpot sizes and rollover rules can vary and the smaller prizes also affect the potential return.

Here we're only looking at some of the jackpot chances, and not the potential returns.

TERRITORY

GAME

JACKPOT CHANCES

HIGHEST PRIZE

Poland

Mini Lotto

1 / 850,668

€ 700,000

New Zealand

NZ Lotto

1 / 3,838,380

NZ$ 44,000,000

Sweden

Swedish Lotto

1 / 6,724,520

€ 25,600,000

Austria

Austrian Lotto

1 / 8,145,060

€ 9,600,000

Ireland

Irish Lotto

1 / 10,737,573

€ 19,000,000

France

French Lotto

1 / 19,068,840

€ 23,900,000

Spain

Spanish Lotto

1 / 31,625,100

€ 33,400,000

UK

National Lottery

1 / 45,057,474

66,100,000

Australia

Powerball

1 / 134,490,400

AU$ 107,000,000

Europe

EuroMillions

1 / 139,868,160

€ 190,000,000

USA

Powerball

1 / 292,201,338

$ 1,586,500,000

Italy

SuperEnaLotto

1 / 622,614,630

€ 177,700,000


SUMMARY of NUMBERS!
How many numbers matched How many ways you can do it
6
1
5 + bonus
6
5 no bonus
312
4
20,670
3
468,520
2
4,392,375
1
17,218,110
0
22,957,480
Total ways = 45,057,474

There are more stats and facts about games here:

Craps - How to play, Chances and Payouts

Blackjack - Hit, Stand, Double Down or Split?

Chances of Card Hands

The Betting Guide - and how to work out the Bookie's profit!

The Roulette Wheel ... and how to worry the casino!

The One-Armed Bandit Slot Machine

How many ways can you put a pack of cards in order?

The National Lottery Odds

Kjartan's Games Home Page

Contact/Enquiries


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