We often find bookmakers offering combination bets such as “over 1 goal, over 3 corner and over 1 card in each half”, particularly in the #YourOdds / #RequestABet / #PickYourPunt type markets. In order that we can benchmark these accurately we need models for each element. This page deals with the cards part.

Benchmarking

The spreads sites are good benchmarks for cards bets. As they offer a buy and sell side of each bet we can use the mid-point as the mean and once we have that use Poisson to calculate the odds of the particular bet we are interested in. The problem we have is the spreads sites quote their prices in “booking points” and we will often need to translate this into number of cards. They award 10 points for a yellow and 25 for a red and this is capped at 35 where a player receives two yellows and then a red.

Not all cards are equal

To make things worse certain bookmakers have different rules for what counts as a card. William Hill for example count a yellow card as one and a straight red card as one but if you get a yellow and then a straight red that counts as two and if you get a two yellows and then a red they cap it at two cards. Bet 365 are the same apart from where a player receives a straight red, they count this as two cards.

The solution

We have looked at 12,984 games in UK leagues over the past 8 years and found that there were 42,326 cards of which 40,185 were yellow and 2,141 were red. On average there are 3.0950 yellow cards per game and 0.1649 reds. We can use these figures to come up with a standard coefficient for converting to cards from booking points but we also want to take into account the increased influence those red cards can have in the more violent games.

We split the data and analysed the ratio between the two card types against the total number of cards/booking points. The results of this are shown in the graph below.

The fits we have give an r squared of 97% which is enough to be confident in the equations produced. We can therefore use these to give a reliable conversion of booking points to cards for all but the most vicious of games.

If we come across a game with more than 90 expected booking points we will have to consider these as outlier games and be careful with whatever benchmarking we are carrying out.