The chances of winning the
UK National Lotto
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The sixth Murderous Maths book "Do You Feel Lucky?" explains that the chance of matching all six numbers in the UK's National Lottery draw is 1 in 13,983,816. However you can also win prizes if you match three, four, five or "five plus the bonus ball".
Some of the calculations which follow can be a little complicated but basically here are the approximate chances of all the different prizes:
-
JACKPOT (matching all six numbers): 1 in 14 million
-
FIVE PLUS THE BONUS: 1 in 2.3 million
-
FIVE: 1 in 55,000
-
FOUR: 1 in 1,000
-
THREE: 1 in 57
And just for interest, here are some other chances:
-
The chance of winning ANY prize: 1 in 54 (about 2%)
-
The chance of only matching 2 numbers: 1 in 7.5 (about 13%)
-
The chance of only matching 1 number: 1 in 2.4 (about 41%)
-
The chance of not matching ANY numbers: 1 in 2.3 (about 44%)
-
How to calculate the different chances
The main thing to know is that the total number of ways of choosing your six numbers is:
JACKPOT CHANCE: only ONE of these ways will win you the jackpot, so the jackpot chance is 1 in 13,983,816 or about 1 in 14 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 43 numbers that are not drawn, so the number of combinations of 3 can you choose from 43 is...
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 12341 = 246,820. That's how many ways you can match three winning numbers, so your chances of matching 3 numbers are 246,820 in 13,983,816. Obviously this looks a bit mind boggling, so to make it simpler you divide 13,983,816 by 246,820 to work out that the chance of matching 3 numbers is roughly 1 in 57
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 43 losers. The sums look like this:
As there are 13,545 ways of matching four
winning numbers, your chances are 13,545 in 13,983,816. It works out that
the chance of matching 4 numbers is roughly 1 in 1,032
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 43 losers. (This last bit's easy: if you have a choice of 43 things and you can choose one, how many choices have you got? 43 of course!) Here's the sums:
This means there are 258 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 258 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 49, there are there are 43 numbers left that the bonus can be chosen from, so it's 1 chance in 43.
We've already worked out that there are 258 combinations that match five balls, and 1 in 43 of these combinations will have the sixth ball match the bonus. Therefore the number of combinations that match five numbers PLUS the bonus is 1/43 x 258 = 6.
This leaves 258 -6 = 252 chances that match five numbers but NOT the bonus.
So for just five numbers, the chances are 252 in 13,983,816 which means the chance of matching 5 numbers without the bonus is roughly 1 in 55,491
And for five numbers and the bonus, the chances are 6 in 13,983,816 which means the chance of matching 5 numbers plus the bonus is 1 in 2,330,636 or roughly 1 in 2.3 million
Where does
13,983,816 come from?
The way the lottery works is that you have a card with the numbers 1-49 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 49 available.
In maths we
say "how many different combinations of 6 can we choose from a total of
49" 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
the book.)
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 49x48x47x46.... right down to ...4x3x2x1. The bottom line is 6x5x4x3x2x1 x 43x42x41x40...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 13,983,816 combinations in total, and only ONE of them will win you the jackpot. So your chances of winning the jackpot are 1 in 13,983,816.
By the way,
in the New Zealand lottery you choose 6 numbers from 1-40 so the total
number of combinations is:
This makes your chances of winning the New Zealand jackpot 1 in 3,838,380,
so you're almost four times as likely to win.
information supplied by www.murderousmaths.co.uk
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And now a slightly simpler explanation !
In the British National Lottery, six ball
are selected at random from forty-nine numbered balls. Players have to guess
which six balls will be drawn. If they get all six correct, they win the
jackpot prize.
We need to work out the total number of
possible ways of choosing six different numbers from forty-nine.
Note that the order of choosing the numbers does not matter.For
example, the numbers 2 , 7 , 34 , 21 , 46 , 11
give the same winning combination as 34 , 11 , 7 , 21 , 2 , 46.
- There are 49 possible choices for the first ball
- For each of these 49 choices of the
first ball,
there are 48 possible choices for the second ball
(because one ball has already been taken out)
meaning there are 49 x 48 ways of choosing the first two balls. - However, because the order doesn't
matter, this has to be divided by 2
(because the second ball can come before or after the first ball)
- So the number of different possible
choices for two balls can be written as
- There are then 47 possible choices for
the third ball,
but since it doesn't matter which position the third ball goes in
we divide by 3, giving
- This sequence continues for all six balls which are chosen.
So the total number of possible ways of
choosing six balls from forty-nine is
which is equal to
13 983 816
or approximately 14 000 000 (14 million).
Therefore, the chance of a single ticket
winning the
jackpot is approximately one-in-14million
USEFUL TIP: If your calculator has a button
marked nCr
you can get the answer directly.
"n" stands for the total number (here n=49)
"r" stands for the number you want to select (here r=6)
So if you type in the sequence 49 nCr 6 =
you should get the result 13 983 816 directly.
nCr is shorthand for the number of different ways of choosing r items from n where the order does not matter.
Information supplied by www.maths-help.co.uk
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