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So, how does the theory go then? It's all to do with combinations of numbers. This page gets a bit complicated so, get a cup of tea and a biscuit or two, then read it. It's all fascinating stuff.....                                                                 

Introduction & Probabilities

In the UK lottery, or Lotto as it is now known, there are 49 numbered balls and 6 opportunities to draw any particular number (let's forget the bonus ball for now). Of course, once a number has been drawn, it cannot be drawn again so, after the first ball is drawn there are only 48 left. Thus the probability in a particular number coming out first is 1 in 49. This probability rises to 1 in 48 in the second ball, up to 1 in 44 for the sixth ball. Every reduction in the available ball-pool increases the probability of a particular remaining number being drawn.

Probabilities are multiplied together when considering all numbers coming up together. So, the probability of (your) 6 numbers matching is given by:

P(%) = 6/49*5/48*4/47*3/46*2/45*1/44*100 = 0.000007%. It is tiny.

Why the 6,5,4,3,2,1? In the first draw, when there are 49 balls, you have 6 opportunities to match one of your numbers (as you have 6 of them!). After the first has been drawn, only 5 are left, and so on.

Combinations

A combination is just that- a collection with no particular regard to the order. So, a tea, biscuit and cake are the same as a cake, tea and biscuit. OR.... the numbers 1,3,5 are the same as 3,1,5 and 5,3,1 etc ^Note. 

Combinations are written ⁿC_r. For Lotto this would be ⁴⁹C_6, the n being the available range and r being the "team size" as it were.

By selecting 6 balls  from 49, there are nearly 14 million combinations of six numbers available. The actual number of ⁴⁹C₆ is 13,983,716.

At 1 draw per week, in a perfectly random situation, it would take 275,000 years to draw every number; most humans do not want to wait that long!

3 Number Evaluation

Clearly there is no possible way to predict which 6 numbers will come up, although many try. To see the effect on the quantity of numbers you have to match to win anything, lets consider 3, rather than 6, as the "full-house", such as

[36 24 18]

The nominal probability of matching 3 or more numbers is about 1 in 54, or just over once a year, mostly defined by the £10 wins - the three balls

You can see that each number you have to get hugely decreases your chance of winning, which is why there are a large number of £10 wins in any draw, less 4-ball, even less 5 balls and sometimes no 6 ball at all. The bonus-ball is also very hard to win, being not much better than than actual jackpot in practice - you still need the first 5 balls!

Whilst the  Lotto is mostly random, the author has picked up the small, but measurable, bias that all systems exhibit and has increased win rates by about 30% - quite significant in reality.

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