h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Hi,

I'm your local neighbourhood betting shop manager.

Here's how to settle all of the bets above, the quick way.

Firstly, ignore your losing selections, as you get nothing back from those, so we only have to consider the winning selections.

Each of your winners has a price, such as 3/1. Consider this as a fraction, and calculate it's value in decimal. For instance 3/1 = 3.00, 7/4 = 1.75, etc. Your returns from single bets on each of your selections would be this fraction plus one, i.e. 3/1 = 3.00 + 1 = 4.00, and 7/4 = 1.75 + 1 = 2.75. Evens is 1/1, = 1.0 + 1 = 2.00

Work out the single returns for each of your selections as above. (Price as fraction plus one)

Where you have non-runners, use the value 1.00

Add an additional one to each of these numbers, and multiply these numbers together.

Subtract one from the result.

If your bet did not include singles, subtract each of the single returns (i.e. the numbers you worked out after adding one, but before adding the additional one).

Multiply the total you now have by your stake unit in pounds.

A worked example:
A 20p win yankee with four winners, at 3/1, 7/4, 5/1 and 13/2.
The first stage yeilds the values 4.00, 2.75, 6.00, and 7.50
Multiplying these together gives 495.00
Subtract one, which leaves 494.00
The yankee doesn't contain singles, so remove those (subtract 4.00, 2.75, 6.00 and 7.50)
This leaves you 474.75
Multiply by your stake, 20p, or 0.20 pounds, and you have £94.95

For each way bets, you need to do this calculation twice, once for winners, and once for the place part of the bet, so you need to know which selections were placed at 1/5 or 1/4 odds. You also need to take into account any dead heats or rule four deductions before you add the additional one. This is a topic for another day, methinks.

Why this works, for the mathematically curious..

It's all down to what happens when you multiply out equations of the form (X+1) x (Y+1), like your maths teacher made you do at school, despite there being no obvious use for it. When you multiply out that equation, you get XY + X + Y + 1. If X and Y are the returns from single bets, then XY + X + Y + 1 is the return from the two singles and the one double on those selections, plus an additional one, which is why we subtract one at the end of the multiplication.

Similarly, (X+1) x (Y+1) x (Z+1) = XYZ + XY + XZ + YZ + X + Y + Z + 1

All of this algebra and mathematical drivel will be the subject of a new article I'll be putting together soon. If you'd like to help, or if you have specific questions, pop along over to A982389 and say Hi!

how do you work out 5folds 6folds in 10 selections

Sorry for the delay, I've been rather busy.

Anyway, the bad news is the method for doing this by hand is not as easy as you might have hoped, but it's not too bad.

Firstly, throw away all of your losing selections. Assuming that somehow you have ten winners left, first of all it's your round, and secondly, get a sheet of paper and turn it ninety degrees so that it's wider than it is tall.

Down the first column, write down the prices of your winning selections. Next to those prices, write the basic factors associated with those selections, adjusting as necessary for rule-4's, dead heats, etc. We'll call that column SINGLES.

Right, listen very carefully, I will say this only once...

To fill in Column X, add up all of the figures in column X-1 that are ABOVE that figure, and then multiply the total by figure in the SINGLES column on that line. Because there are no figures above the first line in the SINGLES column, the first column in the next column, which we'll call DOUBLES is blank.

Rinse and repeat for the next column, which we'll call TREBLES.

Now, if you add up all of the figures in the SINGLES column, it's fairly obvious that you'll get the returns, to a one pound unit stake, of singles on your selections. Add up the DOUBLES column, and you get the total for doubles, and so on until you hit your five-folds and six-folds.

There are shortcuts, in that if you work diagonally down and right you save some fingerwork, but I'll save that as an exercise for you to work out.

Just because I'm feeling nice today (as I have two days off before I get to go and take some bets in the North Stand at Fratton Park on Monday night down in the 'Shed of Dread') there is a worked example at A3137366 although I only put in six prices, because I don't plan on typing all night!

Now I actually come to read your question (DOH!), if you are looking for how to STAKE these bets (I.e. how many X are there in Y) then all you need to know is this:

The formula for how many X in Y is Y! / (X! x (Y-X)!), also known as nCr if you have a scientific calculator (which is not recommended for bet settling by the way, see my article), in which case you would hit 10 nCr 6 and get the answer you are looking for.

Y! is the 'factorial' of Y, which is what you get if you multiply Y by all of the smaller integer (whole) numbers less than it. 5! is 5x4x3x2x1 which is 120.

A shorter way of working it out (actually, exactly the same way, just a bit optimised) is to start with the larger number, and multiply that by progressively smaller numbers until you have multiplied the 'smaller number' number of numbers. Then start dividing by the smaller number, and each number smaller than it, until you hit 1.

So, sixfolds from ten:

10 x 9 x 8 x 7 x 6 x 5

/ 6 / 5 / 4 / 3 / 2 / 1

So for ten selections there are 210 sixfolds, and 252 fivefolds.

Another way to work this out is to grab a copy of Pascal's Triangle. Details at A280306. The first row represents zero selections, the second represents one selection, the third two selections, and so on. In each row, the first number is always one, and is the number of zero-folds in that bet. (Not entirely useful ) The next number is the number of singles, then doubles, and so on until you end up back at one, which represents the accumulator. You'll notice the 4 6 4 1 pattern, which is the distribution of bets in a lucky 15 with four selections.

You'll find a table on the back of our football coupons ('our' being the bookmakers beginning with 'L' in your local high street, which, this being the beeb, I couldn't possibly mention by name, eh?).

i had a double up today on a 50p round robin total bet =£5.00 the winners were 7/1 and 14/1 thanks

Assuming win only and no rule 4's or other sillyness
You have two singles on each winner, with one stake deducted to fund any any-to-come single on each of the others. As there are two winners, you have those two singles, plus one additional single (with no deduction) on the other two.
Thus: 7/1 and 14/1, basic factors 8 and 15, three singles = 3 x (8+15) = 69, less four stakes = 65.
The round robin also includes a trixie (doubles and treble), so you have a winning double of 8 x 15 = 120.
Total returns should have been 120 + 65 = 185 stakes, or £92.50
(I'm doing this in my head here, so I may be utterly wrong)
Had you just done a £1.25 trixie (also £5 total) you would have received £150

There are easier ways of settling bets of this genre (round robin, flag, super flag (basically combination single-stakes-about + all multiples), but they give me a headache. But if you're really desperate, I'll write them up.
Personally, the round robin only provides value when you are looking at a single winner, and even then, it's hardly worthwhile if backing anything that is at a low price, especially odds-on. Backing unnamed favourites, or anything <9/1 each-way is crazy with RR's. Your and other's opinon many vary.

Standard disclaimer applies. Back in the days of betting tax I'd always say "If you think your selection is going to win, pay your tax. If you don't think your selection is going to win, don't bet on it"