Lottery Probability.

Hi.

The UK National Lottery is based on 49 balls.

6 balls are drawn for the main part and 1 additional bonus ball (from the remaining 43 balls) is also drawn.

On each ticket, you choose 6 of the 49 balls.

The bonus ball acts like a joker (but has less value) when you have matched 5 of the drawn numbers.

The odds of matching all 6 balls is 1 chance in 13,983,816. (What's the exact mathematics for this?)

The odds of matching any 3 main balls is 1 chance in 56.66. (Again, what's the maths?)

So. With that, take my lottery wheel.

We have chosen 7 numbers (at the beginning of last year, it was the top 7 numbers drawn - more on this later).

With that, we have 7 tickets with each number represented 6 times.

How do I now calculate the odds of winning?

If I match 3 numbers (56.66 chance), I'll win with 4 tickets.
If I match 4 numbers (1,032.4 chance), I'll win with 3 tickets of 4 numbers AND 4 tickets of 3 numbers.
If I match 5 numbers (54,200.84 chance), I'll win with 2 tickets of 5 numbers AND 5 tickets of 4 numbers.
If I match 6 numbers (13,983,816 chance), I'll win with 1 jackpot of 6 numbers AND 5 tickets of 5 numbers.
If I match 6 numbers AND the bonus ball, I'll win with 1 jackpot of 6 numbers AND 5 tickets of 5 numbers AND the bonus ball.

But this is NOT based on any particular pattern or other information.

I started the wheel 2 years ago. Since then only 2 numbers have dropped out of the top 7 numbers drawn.

http://www.flight1721.co.uk/lottery.php with numbers 38,44,31,25,23,32,43.

Admittedly, it seems that Wednesday is a better day so far than Saturday in number of wins.

Is it possible to take into account not the chances of winning a single drawn, but also to take into account the frequency of the numbers having been previously drawn?

So, whilst number 38 has been drawn 138 times (the most popular ball), if it has not been drawn for the last 100 draws, can we account for this?

So. To recap.

1 - What is the mathematics behind the 13,983,816 to 1 chance or 6 balls from 49?
2 - The same for 3 balls from 49.
3 - What effect does having more tickets have when the numbers are wheeled?
4 - Can past performance be taken into account?

NOTE: I know that there is absolutely no way that any mathematics can guarantee a win. What I want is to be able to provide some better statistics to this page than I already have.

I am good enough for my job at mathematics, but no way near anything useful for this sort of maths, so please keep things simple.

Regards,

Richard Quadling.

P.S. Unless someone can come up with ALL the answers, I am expecting to split these points.