THE STAT MAN

Chapter 10 · Worked solutions

Chapter 10 workings: The Acca Tax

Chapter ten worked: how leg margins compound in doubles and six-folds, the Craggs check, a value treble, and how many legs take the margin past half.

An accumulator's booksum is the product of its legs' booksums, so margin compounds. The effective margin is 1 − 1 ÷ (combined booksum): the share of the stake the firm expects to keep.

The exercises

  1. A double from two 104.5 per cent legs
  2. A six-fold at 106 per cent a leg
  3. The Craggs check
  4. A treble of genuine 50 per cent shots at 11/10
  5. How many 5 per cent legs pass a 50 per cent margin?

A double from two 104.5 per cent legs

Find the effective margin of a double whose two legs each have a 104.50 per cent booksum.

  1. Multiply

    1.045 × 1.045 = 1.0920.

  2. Convert

    1 − 1 ÷ 1.0920 = 8.43 per cent, against 4.31 per cent for one leg.

The answer

8.43 per cent, close to double a single.

A six-fold at 106 per cent a leg

Six legs, each with a 106.00 per cent booksum. Effective margin of the six-fold?

  1. Compound

    1.066 = 1.4185.

  2. Convert

    1 − 1 ÷ 1.4185 = 29.50 per cent: about 70.5p back per pound on average.

The answer

29.50 per cent. Do it for your own coupon next; the figure will differ.

Open this in the margin remover

The Craggs check

What does 50p return at 2,000,000/1, and what odds would a £1.4 million return need?

  1. Return

    0.50 × 2,000,001 = £1,000,000.50.

  2. Required odds

    £1,400,000 ÷ 0.50 − 1 = 2,799,999/1, near enough 2,800,000/1.

The answer

£1,000,000.50; about 2,800,000/1. The two famous figures do not fit together.

A treble of genuine 50 per cent shots at 11/10

Three legs, each a true 50 per cent chance priced at 11/10. EV per £1 as a single and as a treble, and how often the treble loses.

  1. Single

    0.5 × 2.1 − 1 = +5p.

  2. Treble

    Win chance 0.5³ = 0.125, payout 2.1³ = 9.261. EV = 0.125 × 9.261 − 1 = +15.76p, which is 1.05³ − 1.

  3. Loses

    1 − 0.125 = 87.5 per cent of the time.

The answer

Single +5p, treble +15.76p; the treble loses 87.5 per cent of the time.

How many 5 per cent legs pass a 50 per cent margin?

With a 5 per cent overround on every leg, how many legs before the effective margin exceeds 50 per cent?

  1. The condition

    1 − 1 ÷ 1.05n > 0.5 means 1.05n > 2.

  2. Try values

    1.0514 = 1.9799, not enough; 1.0515 = 2.0789, margin 51.90 per cent.

The answer

15 legs.

These are the workings. The argument for them, and the exercises themselves, are in chapter 10 of the book, The Acca Tax.

Get The Stat Man All worked solutions

Out now · free to read on Kindle Unlimited

Take the mask off the price

Start here Kindle Free to read with Kindle Unlimited Reflowable · searchable · dark-mode friendly ASIN B0GZ4X3L9R Buy on Amazon Paperback The everyday reading copy 6 x 9 in · 571 pages ISBN 9798189116001 Buy on Amazon Hardback Built to be a reference for years Case laminate · 547 pages · lies flat on a desk ISBN 9798189121197 Buy on Amazon