How well did you do?
Did you choose 7 as your number between 1 and 10? this is most people's favourite choice. The
next most popular choice is 3.
Both these numbers are comfortably far in from the extremes without being too close to the
middle. Also they are prime numbers which human beings seem to like. If the choice is truly
random then any number from 1 to 10 is equally likely to be chosen.
The numbers for week 2 were actually 2,3,21,22,23 and 40 so giving this table.
| Week 1 |
1 |
4 |
20 |
31 |
41 |
43 |
| Week 2 |
2 |
3 |
21 |
22 |
23 |
40 |
| Week 3 |
28 |
34 |
41 |
45 |
46 |
49 |
It is very unlikely that you chose these exact numbers but did you spread your numbers out
more evenly than the actual draw? People sometimes feel that having three consecutive numbers
is not very 'random'.
Whatever you guessed
was just as likely to be correct as the randomly chosen lottery results for
that week. The lottery mechanism has no memory of what it chose in previous
weeks, and there is no strategy for working out from previous draws
which numbers are most likely to appear.
You can see too
that numbers chosen in a random way sometimes do come clumped
together.
The same kind of effect can sometimes explain apparently sinister
clusters of illnesses such as childhood leukaemia.
The winning numbers for each of the three weeks in the table have been arranged in
order for convenience but the lottery mechanism won't choose them this way. And the winners
can have thought of their choices in any order. All that matters is that they have the right
six numbers. So how many ways do you think there are that the choice of the six winning
numbers can be made? If you don't know the maths to work this out, try guessing.
There are 6 ways to make the first choice among the 6 winning numbers.
With each of these there are 5 ways of making the second choice.
Then there are 4 ways of making the third choice.
And there are 3 ways of making the fourth choice.
And 2 ways of making the fifth choice.
And just one way of making the sixth choice!
So there is a total of 6 x 5 x 4 x 3 x 2 x 1 = 720 choices for the correct numbers.
Mathematicians call this number 6 factorial and write it as 6! (6 followed by an exclamation
mark).
In a similar way we can see that there is a massive 49 x 48 x 47 x 46 x 45 x 44 =
10,068,347,000 ways of picking 6 numbers from 49 numbers if we count every choice made in
a different order as a different choice. This huge number tells us the total number of
different permutations or arrangements of 6 items from 49 items.
But we've seen that, for any choice of 6 particular numbers, there are 720 possible
different orders of choice. So the total number of choices of 6 numbers from the
numbers 1 to 49 independent of their ordering is
10,068,347,000 divided by 720 = 13,983,816.
This number tells us the number of combinations
or selections of 6 items from 49 items.
Only one of these choices is totally correct so the probability or chance of getting the
right choice of numbers in this lottery is 1/13,983,816 or about 1 in 14 million.
If the chance of being correct is so tiny what do you think the chance is that a lottery
entry will be completely wrong?
There are 43 ways to choose the first wrong number, and 42 for the second one and so on.
The number of completely wrong choices for all 6 numbers independent of order is
43 x 42 x 41 x 40 x 39 x 38 divided by 720 which is 6,096,454.
So the probability of being completely wrong is 6,096,454 divided by 13,983,816 which is
0.436 to 3 d.p. so it is less than half. In other words, people are more likely than not
to choose at least one correct number in any go at a lottery like this. This could make
them feel overly optimistic about their future chance of getting all 6 numbers correct!
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