In one of our rare statistical analysis posts, we explore how two firms (Paddy Power and Betfred) can price up a “black swan” event, and they come no more unlikely than a Javier Hernandez hat-trick for the Hammers v Huddersfield game, priced as disparately as 60/1 (PP) and 25/1 (BF) respectively.
Warning: This will get mathematical, but nothing worse than dusting off that “O” level…
The key basis probability is the chance of a West Ham goal and the chance of Javier scoring it. That work is shortcutted for us by the omnipresent “Anytime Goalscorer” market – which prices a Chicarito strike at 7/5 for PP and Evens for Betfred. Apparently quite closely priced – ignoring over-round markets, that’s a 50% chance per Betfred and 41.7% chance per PP.
However extending that to three goals opens up a fat wedge. To compute three goals from “Anytime”, we first derive “Notime” – the probability of not scoring at all in a game being (1 – anytime). Then, to simplify, we can divide a game into 90 discreet minutes. The chance of not scoring in any given minute is then the 90th root of the Notime probability (i.e. no goal per minute multiplied 90 times over equals no goal per game). For PP (1-0.417)^(1/90) = 99.4% chance of Javier not scoring in any given minute. For Betfred this is 99.2%. The probability of him scoring a goal in any given minute is one minus this probability (0.6% and 0.8% respectively).
The chance of one Javier goal exactly in the game is the probability of no goal multiplied through 89 times multiplied by the probability of a goal in a minute, all multiplied by the number of correct combinations (this is where you dust off those factorials), for one goal that is 90 combinations. We compute one goal probabilities as 31.5% (PP) and 34.8% (BF) respectively.
The chance of two goals is no goal per minute^88 multiplied by a goal per minute^2 multiplied by the number of combinations (90*89/2*1). Being 5.9% and 7.7% respectively.
Finally 3 or more goals is the anytime percentage minus the computed percentages for one goal and two goals. Which computes as 1.7% and 3.2% respectively, representing odds of 58/1 and 30/1 respectively.
Hence a minor gap in original goal expectation explodes into a wide wedge when taken to extreme events.
This is of course rendered pointless should there be a 1-0 Huddersfield win, but if you fancy the Hammers, then “buying small options”, i.e. low stakes on unlikely outcomes like a Javier hat-trick, is one fun way to be at the right end of the odds.
Well done for reading this far, the good news is…no homework.