Recently I read a short blog post by Stephen Kattner about yellow cards. He used a quick calculation to estimate that players in Europe who were carrying yellow cards were 14 times more likely to be sent off than players who weren’t. It turns out the calculation of this statistic is rather more complex, so I asked my colleague Jimmy Coverdale for some data and began crunching the numbers.

There were 1,193 first yellows cards issued in the English Premier League during the 2012-13 season. With 380 games, that’s an average of about 3.1 first yellows per game. Overall, the chance of seeing a player receive his first yellow in a given minute of the game was 3.3%. If we divide this number by the 22 players on the field, we get a back-of-the-envelope estimate of the chance a given player will receive his first yellow in any given minute: 0.15%. Of course, for some minutes there are fewer than 22 players on the field, and some players may already have a first yellow; as a result, our estimate is biased slightly downward. But 0.15% works as a rough approximation.

If we take a closer look, we can see that the chance of receiving a first yellow changes over the course of the game. Early yellows are relatively rare, and the probability increases as the game progresses. As Jimmy points out, this is likely due to time-wasting or the accumulation of fouls (and perhaps of bad blood as well). The histogram below shows the distribution of first yellow cards by minute:

Another way of looking at these data is to consider how many minutes a player can still be on the field after a first yellow, barring a red card or a substitution. Here is a cumulative distribution of first yellow cards by the possible minutes remaining when they were issued:

This graph is back-to-front relative to the previous one. The slope is initially quite steep, indicating that players receive a lot of first yellows in the later stages of the game. It flattens out as we get closer to 90 possible minutes remaining; as we already know, fewer players receive cards early in the game.

As an aside, away teams had 56% of the first yellow cards in the Premier League last season. Red cards, by contrast, were evenly split between home and away teams: 18 straight reds and eight second yellows for each. This could indicate greater aggression by players working in a hostile atmosphere. Alternatively, it could show that home fans are more able to sway referees on minor decisions than on major ones.

But let’s go back to the main question of how frequently players with a first yellow receive a second one. The important insight here is that these players have only a limited amount of time to pick up a second yellow. If the players stayed on the field forever, we’d find out just how long it took for all of them to get that second card. Since they don’t, we have to treat the vast majority of our observations – that is, all the players who received only one yellow card – as truncated or, as statisticians would say, censored.

A classic comparison to this situation, usually called a “survival model”, is the time it takes for light bulbs to burn out. If we have a big box of light bulbs and we plug them all in for 2,000 hours, not all of them may burn out in the allotted time. But we can’t run the experiment forever, so we have to use statistical methods to infer what would have happened if we had.

In this case, the cumulative probability of receiving a second yellow, depending on the number of minutes played after the first yellow, is given by the estimates in the following graph:

There are several ways of interpreting this graph. Perhaps the easiest is to say that Y% of players won’t make it X minutes after their first card without receiving a second. The graph ends at 95 minutes, at which point 3.8% of the players carrying a first yellow have received a second yellow. The probability isn’t completely linear; by 48 minutes after the first yellow (equivalent to about half a game), just 1.5% of the players would have been expected to receive a second yellow.

We can find the probability of receiving a second yellow in any given minute by looking at a “smoothed hazard estimate”, which tries to create a trend to match the underlying probabilities:

Again, the chance of a second yellow is much lower than the chance of a first, with the former ranging from about 0.04% to 0.06%. Even with 60 minutes under his belt after receiving his first card – when the game is probably close to ending – a player carrying a yellow is still less than half as likely to receive another one as other players are to receive their first.

Part of this difference may be due to the caution of managers. They tend to pull off players who seem too aggressive or are too valuable to be lost to suspension; these players never get the chance to pick up a second card. Another part of the difference may be due to players’ own self-control; with one yellow already used up, they’re extra careful.

All of that said, a player on a yellow is still far more likely to be sent off than a player without a card. The overall chance of a straight red in any given minute was about 0.0045% last season, whereas the chance of a second yellow was about ten times as large.

Is this gap between two tiny probabilities enough to make up a manager’s mind? Let’s say his star striker has picked up a yellow card in the 70th minute with his team trailing 0-1. The chance that the striker will end up with a second yellow before the game is over is about 1%. If the team’s next game is the FA Cup Final, then the manager might not want to take even such a small chance. But under any other circumstances, why not?