## Friday, 16 March 2012

### Maths numpty requires help

I'm not very good with numbers. I never was. I failed my maths O Level and, to quote a very old joke, I can just about count to 21 if I'm naked.

Saying that, I can do the basic stuff; the adding and subtracting, multiplying and dividing. And I can work out fractions and percentages. But anything beyond that is a world of confusion and befuddlement for me. Which is why, when trying to work something out today, my brain got all hornswoggled. Here's my problem:

There are 36 possible combinations of numbers when you throw two dice:

1+1, 1+2, 1+3, 1+4, 1+5, 1+6
2+1, 2+2, 2+3, 2+4, 2+5, 2+6
3+1,3+2, 3+3, 3+4, 3+5, 3+6
4+1, 4+2, 4+3, 4+4, 4+5, 4+6
5+1, 5+2, 5+3, 5+4, 5+5, 5+6
6+1, 6+2, 6+3, 6+4, 6+5, 6+6

Mathematically, these are all different results because Dice A and Dice B could both land in one of six positions so there are 6x6 possible outcomes = 36. However, if we look at it purely from the point of view of someone playing dice, then there are only 21 numerical outcomes as 5+2 is the same as 2+5 to a gambler;

1+1, 1+2, 1+3, 1+4, 1+5, 1+6
2+2, 2+3, 2+4, 2+5, 2+6
3+3, 3+4, 3+5, 3+6
4+4, 4+5, 4+6
5+5, 5+6
6+6

So, if I wanted to throw a Seven, there are six possible mathematical possibilities:

1+6
2+5
3+4
4+3
5+2
6+1

But if I were a gambler at a casino, 1+6 would be the same as 6+1 as I wouldn't need to worry about which die showed which number as long as the total of both added up to seven. Therefore, there would be just three ways to throw a seven. So here's my question:

Mathematically, if there are six possible ways to get a seven with two dice and there are 36 possible combinations, then I have a 1:6 chance of throwing a seven.

But if I am a gambler in a casino and there are 21 possible combinations of two dice and just three ways of getting a seven, I have a 1:7 chance of throwing a seven.

Why aren't they the same odds?

It's always risky putting up a post like this and showing people just how dim I really am. But I'm never afraid to say 'I don't know' and I'm always willing to learn. So can anyone explain it to me please? Short sentences preferably. Brain. Hurty.

Postscript: Thanks to some very clear explanation by people replying to my blog and my heartfelt plea on Twitter I now understand the difference between mathematical probability and a gambler's odds. It's all a matter of doing the right sums.

It's reminded me of that old puzzle where three people go out for a meal with £10 each. The meal comes to £25 so the waiter puts £25 in the till, pockets £2 and gives £1 change back to each of the three diners. By asking each diner how much their evening has cost them (£9 each) and adding it to the £2 in the waiter's pocket we get a total of £29 and the question 'Where's the missing £1'. But it's based on a false sum just as the 1:7 chances of a gambler getting a seven is based on a false sum. The truth is that there is £25 in the till, £2 in the waiter's pocket and £3 between the diners.

Maths is so cool. I just wish my brain was wired up for it.