All the chances of winning on the UK National LotteryTHE GAME:You select six numbers from 159. Six numbers are drawn plus a seventh "bonus ball".
 
How to calculate the different chancesThe main thing to know is that the total number of ways of choosing your six numbers is:
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.
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:
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.  
 
Where does 45,057,474 come from?The way the UK lottery works is that you have a card with the numbers 159 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.
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
