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I found this information, but I am not find the orginal author.

The information is based around the UK National Lottery but I think most of the information will apply to 49 ball lotteries.

Odds

The jackpot

6 Numbers are drawn at random from 1 to 49. There are 49!/(6!*(49-6)!) combinations of numbers. The order in which the numbers are drawn doesn't matter. The number of choices is 13,983,816, So the odds are about 1 in 14,000,000.

5 plus the bonus ball

You are still matching 6 numbers, but in 6 ways. The odds are 1 in 13,983,816/6 or 1 in 2,330,636

5 Numbers

This is 42 times more likely than getting 5 plus the bonus ball because after the first 6 balls are drawn, there are 43 balls left, and you can match 42 of these 43 balls without matching the bonus ball. The chances are 1 in 2,330,636/42, which is 1 in 55,491.33

4 numbers

Break the problem down into 2 parts. Firstly the odds of matching the first 4 numbers and not the last.

Number = 1 in 49/61

2 Number = 1 in 48/5

3 Number = 1 in 47/4

4 Number = 1 in 46/3

5 Number = 1 in 45/(45-2)

6 Number = 1 in 44/(44-2)

We can compound these together. There are then 15 combinations of 4 from 6. [6!/(4! x (6-4)!)] Result is 1 in 1032.40

3 numbers

Same technique as 4 numbers but there are 3 from 6 combinations. [6!/(3! x (6-3)!)] which is 20. Result is 1 in 52.79 3 numbers

Strategy

Lucky dip.

Number of people using lucky dips are small. The impact of lucky dip on what happens should not be underestimated.

People are bad a picking random numbers but lucky dip changes that. Roll overs have become more infrequent since lucky dip was introduced.

Choosing numbers

If you are playing more than one board then there are strategies that can improve your results. There are two 'results' that you can improve. Firstly, you can have wins more frequently.

Consider two strategies for playing five boards each week.

  • 1. Pick 6 numbers and play 5 boards, all with the same numbers.
  • 2. Pick 30 different numbers and play 5 boards. No numbers will be shared between the different boards.

Strategy two will win 5 times as often as strategy one.

Now consider the return on you bet.

If you win £10 then there is no difference in your return.

However, if you win more that £10, and in particular if you do win a large prize, then for a one pound bet your return is higher using the second strategy.

This is because in the first case doubling your bet does not double how much you win.

If you consider the extreme case, where you are the only winner you win the same amount of money using both strategies, but you have bet 5 times the stake using strategy one to win.

So pick different numbers, in case you win big. If you don't think or want to win big, then don't play the lottery!

This then leads on to some special cases for betting different numbers.

Playing more than eight boards.

How do you choose numbers when you bet more than 8 boards (8 * 6 = 48)? The first eight boards are easy, pick unique numbers.

When you pick more than eight you don't want to pick numbers all from one of the boards on the first pass, so your numbers should mix up the numbers from the first pass.

Consider a simple case of 2 boards of six.

First pass gives

(1,2,3,4,5,6) and (7,8,9,10,11,12)

A good choice for the second pass would be

(1,2,3,7,8,9) and (4,5,6,10,11,12)

(For clarity I have omitted using random numbers)

This is similar to generating Snobol or Fauvre number sequences for Monte Carlo simulations.

This is an interesting result because it is the complete opposite from the football pool idea of a permutation.

That is picking a small set of numbers and then picking all combinations of those numbers.

How good are people at picking numbers?

The short answer is very bad. About 65 million bets are placed each week. There are approximately 14 million choices.

There is a roll over every 5 to 6 weeks. When there is a roll over that means we have a hole in choices made by players.

Now a quick program suggests that for 14,000,000 possible choices, 65,000,000 random selections there would be about 125,608 holes (at least on this simulation).

This would imply a roll over of 1 in 111 games!

Generating Random Numbers

This is the main reference to start with to learn how to generate good random numbers. Knuth, D.E., 1981; The Art of Computer Programming, Volume 2 Seminumerical Algorithms, Addison-Wesley, Reading Mass., 688 pages, ISBN 0-201-03822-6 Press, W.H., B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, 1986; Numerical Recipes, The Art of Scientific Computing, Cambridge University Press, Cambridge, 818 pages, ISBN 0-512-30811-9

Which numbers do people use?

This is of real importance. If you win a large prize, you want to be the only winner. I worked with someone who won the jackpot.

Luckily he didn't find out until the Monday when it was know there were 133 winners or I can't imagine what he would have told his boss about his job from the pub where he was celebrating with his friends!

So you want to pick numbers, or combinations of numbers that other people don't.

Camelot will not tell you what the most popular and unpopular numbers are, and I think that the introduction of lucky dip has changed things.

However, it is possible to make some very good guesses. If we look at the information that is available, we have 7 numbers, the number of winners in each of the categories, and the total number of bets made.

First, consider just the £10 wins.

What can you say if there are a large number of £10 wins compared to what could be expected using an even selection of numbers. My conclusion is that some of the 6 numbers that came up must be popular numbers.

So I assign a score for each week depending on the number of £10 wins. I give each of the winning numbers that week that score.

I repeat this for multiple weeks, and average the scores for each number, every time it comes up.

I can now sort the numbers according to their score, the most popular numbers being those with the highest score.

In fact this is a simplistic solution. Other winners should affect the score. I take this into account giving £10 winners a weight of 3/6, 4 ball wins a weight of 4/6, 5 and 5 plus bonus a weight of 5/6 and jackpot winners a weight of 6/6. I ignore the bonus ball as you cannot tell anything from it.

This is useful because you want to bias you numbers to the unpopular. The most popular numbers are 7, 17 ... I'm not going to tell you the unpopular numbers, because of self interest.

If I win the jackpot, you will know because you will find the list here then!

How do people pick there numbers?

Looking at the distribution there is a bias towards the top right of a board. I have a conjecture that this is due to the majority of the population being right handed.

Further work is needed on pairs.

Do people avoid pairs of consecutive numbers? What about consecutive numbers that appear on different lines on a board? I'm still thinking about this.

I suspect the first is true, but not true for the second cases.

The accepted theory is that low numbers are picked more often, because people use house numbers, birth dates and there are more houses numbered one than any others.

Also using birth dates bias 1 to 12 and 1 to 31 more than others. Whilst there is probably an element of truth in this I think the reason for the bias to low numbers is simpler.

People start picking numbers from the low to the high. When they get to the high numbers usually they have already picked 6 numbers. High numbers just don't get a look in.

Should you pick the same numbers each week?

Psychologically is may be bad for you.

If you stop playing, or miss a week and your numbers come up, could you cope.

Is there then any rationale in picking new sets of numbers each week. I think there is.

If you pick a set of numbers, you don't know how good a set they are until you win, you can just make guesses.

So if you stick with one set, you could be lucky and pick good sets, or unlucky.

If you change your sets regularly, then you can say that on average you will be picking average sets.

Diversification helps.
 
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