Introduction To Probability - 1.5
June 16th, 2009 by reddavis
How would be go about calculating the probability of winning the lottery?
In total there are 7 balls that are drawn for the UK National Lottery: 6 normal balls a bonus ball. The seven balls are all drawn from the same batch of 49 balls.
To receive first place you need all 6 normal balls. To understand whats going on here we need to understand factorials.
Factorials
There are 3 people in a room: A, B, C. If we were to ask them to leave the room, what are all possible combinations in which they leave the room:
ABC
ACB
BAC
BCA
CAB
CBA
A way to calculate this is: N = 3×2x1
Now if there were 24 people, the number of combinations would be: N = 24×23x22… Now this equals a rather large number, which would make writing calculation a pain in the ass, so we shorthand them like so:
24!
This is a factorial.
Let’s say we now have 4 people in a room and ask them to leave just like we did before, but this time we only care about the first two people who leave. How many different combination can we have? We can work this out like so:
4×3 = 12
If we had 24 people and only cared about the first 5 we would calculate it by:
24×23x22×21x20
As you can see, it can get a bit boring having to type all of that into a calculator, so we can use this instead:
n! / (n-r)! where n = number of objects, r = amount we care about
From this we can work out that:
C = 4! / (4-2)!
C = 24 / 2
12 Different combinations
Back to the lottery…
You may noticed that the method I was just talking about care about order but in the lottery it doesn’t matter if the number 2 comes first or third (disregarding the bonus ball as we have no care in it to win). To disregard order we can calculate the number of ways 6 numbers can arranged:
6! = 720
We take the total number of combinations for 6 balls and divide it by this number:
49×48x47×46x45×44 / 720 = 13,983,816 combinations
This means that the probability of us winning first price is 1/13,983,816
This looks messy again, so lets combine this with what we already know:
Number of Combinations - Order Matters
S = n! / (n-r)!
Number of Combinations - No Order
T = S / r!
And this is turned into:
nCr = n! / (n-r)! x r!
Next time we’ll calculate probability of winning other place prizes
































