A Cornish Patsy for Breakfast

A Cornish Patsy for Breakfast

Is there a typo here? Did Ptarmigan Todd mean Pasty instead of Patsy? Whichever he meant, the challenge will be to write something coherent combining Patsy/Pasty with Paul the Octopus.

Paul the Octopus. Pundit Supreme or Lucky Boy?

Life often presents us with yes or no type decisions. During the 2010 Football (Soccer) World Cup this applied to octopus life too - offered two alternatives (the two competing teams) Paul the Octopus chose the winners 12 times out of 14. Was this apparent excellence just a case of getting lucky or did he really know what he was doing when predicting the outcome of football matches?

If he was simply lucky then the situation can be likened to tossing a fair coin - heads or tails for us British folk though Ptarmigan is conscious that somewhere in the world there could be a coin so thick that there is a likelihood of it landing on its edge. Luckily, Ptarmigan does not live there so let’s stick to an experiment of tossing a British coin 14 times, the outcome of each being an equal chance of a head or a tail.

You could spend a month of 8 hour days tossing a two pence piece and recording each result (don’t send them to Ptarmigan) but it’s usually quicker to use the following formula …

N! * 2^-N / (H! * (N - H)!) where

N is the number of coin tosses

H the number of heads required

(N-H) the number of non-heads (tails) required

An exclamation mark behind a number means it is to be taken as a factorial which is shorthand for saying take it backwards from here in steps of one. So 14! = 14 * 13 * 12 * and so on down to 3 * 2 * 1. Luckily it can be cancelled out by other factorials, for example 14! / 12! just boils down to 14 * 13. So, what’s the probability of Paul getting lucky 12 times out of 14? Using the formula above …

14! * 2^-14 / (12! * 2!) = 91 * 2^-14 = .00555 to five decimal places, or if you prefer to think in percentages, just move the decimal place two to the right to give 0.555%. Or, to put it another way … someone with no interest in or knowledge of football has a bit over one half of one percent chance of simply guessing the winning team 12 times out of 14 matches yet to be played. It’s an unlikely outcome but certainly not impossible.

For 14 tosses, a complete table of outcomes and their probabilities is

Heads Tails Probability (5 dec places)

0 14 .00006

1 13 .00085

2 12 .00555

3 11 .02222

4 10 .06110

5 9 .12219

6 8 .18329

7 7 .20947

8 6 .18329

9 5 .12219

10 4 .06110

11 3 .02222

12 2 .00555

13 1 .00085

14 0 .00006

The total probability in this table should add to 1 but due to rounding to 5 decimal places it only adds to .99999 but that’s close enough.

If you can be bothered to graph or histogram the results you’ll find a nice symmetrical distribution which (hopefully) matches common sense where the most likely outcome is 7 heads and 7 tails, whereas the 14-0 or 0-14 outcome is very, very unlikely at one six thousandth of one percent - if you get one of these results it might be sensible to ask, is this a fair coin?

So, Paul, football pundit supreme or lucky octopus? Common sense is probably leading you to the latter but since Paul was a proper west country lad (he was hatched in Weymouth, Dorset) Ptarmigan prefers to think that he must have been a very fine octopus indeed!

So, a coherent combination of Cornish Patsy/Pasty and Paul.is needed. Ptarmigan is rolling up his sleeves as you read this…It’s in writing as they say.