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
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
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
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 numbersSame 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
Lucky dip.Number of people
using lucky dips are small. The impact of lucky dip on what happens should not
People are bad a picking random numbers but lucky dip
changes that. Roll overs have become more infrequent since lucky dip was
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
- 1. Pick 6 numbers and play 5 boards, all with the same
- 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
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
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 NumbersThis
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
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
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
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.
there then any rationale in picking new sets of numbers each week. I think
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.
change your sets regularly, then you can say that on average you will be
picking average sets.