I am reading an article talking about the mathematics in football (link: https://plus.maths.org/content/ball ). In the second part, it asks if a player should risk being sent off in order to gain the opportunity of having a goal, providing that a foul will not lead to penalty kick.
I am not sure about what the writer talks about in the following part, especially the notation of 'crossover time'. Would someone help out me in understanding the writer's claim?
We need to know how likely it is that a goal will be scored if the offence is not committed. This will seldom be an easy judgement to make, but it is crucial to making the right decision. That decision can be neatly described by identifying a crossover time, T, corresponding to the particular chance that a goal will result. A player should risk a red card if, and only if, the crucial moment arises at time T or later. For games at a good professional level, a snapshot of the values is
Chance of a Goal --- Crossover Time
$100\% $ --- 16 minutes
$60\% $ --- 48 minutes
$30\% $ --- 71 minutes
Recall that this table applies when a "professional foul" would not lead to a penalty kick. Even the best players stumble, or mis-hit their shots, so the chance of a goal will only rarely be close to $100\%$. A player who gets sent off in these circumstances before half-time is likely to have made a miscalculation! On the other hand, very late in a tight game, the table suggests that if an attacker has a non-negligible chance of scoring, doing the nasty deed may be best for your team. As a football enthusiast, I dislike this implication of the table. But it suggests the desirability of changing the rules of soccer to encourage fair play: permit a referee to award a penalty kick (or even a goal) if a defender is sent off in these circumstances, wherever the crime is committed!
