A Football Probability Question
This came up in last week’s Broncos-Chargers game.
The Chargers scored a touchdown with a minute and a half left in the game. Broncos were out of timeouts. Before the extra point, the Chargers are up by seven. Question: if you are the Chargers, do you go for two, or kick the extra point? The announcers were debating the issue, and I’ve seen more debate online, but there is actually a mathematically correct answer here.
The arguments were: (pro) if you go for two, and make it, then the game is over. If you don’t, the Broncos still have to drive the length of the field; not likely. (con) Kick the point, then the Broncos have to drive the length of the field – not likely – and then make a two-pointer, also not likely.
Three assumptions. (i) If the Chargers go for two and make it, then the Broncos have a negligible chance of winning. (ii) If the game goes to overtime, each team has a fifty-fifty chance of winning. (iii) The odds of making it when you go for two are less than fifty-fifty.
It’s easy enough to write out some equations that reveal the truth, but there’s an elegant way to argue this. In either case, in order for the Broncos to tie the game, they have to both drive the field and score a touchdown, and there is a two-point conversion that has to go the right way. The only difference is that if the Chargers go for two, they are the ones that have to make the conversion, whereas if they kick the point, the Broncos are the ones to go for two. Since the chances of making the two-point conversion are less than fifty-fifty, the best choice for the Chargers is to force the Broncos to have to make the conversion, i.e. the Chargers should kick the point.
Not sure that’s an especially clear explanation. Here are the formulas. Say the probability of the Broncos driving the field for a touchdown is TD, and the probability of a team converting the two-pointer is C. Then if the Chargers go for two, the Broncos win with probability
and if the Chargers kick it, the Broncos win with probability
Since C < ½, the latter number is going to be smaller, which is what the Chargers want. So they made the right choice – go figure.
The Chargers scored a touchdown with a minute and a half left in the game. Broncos were out of timeouts. Before the extra point, the Chargers are up by seven. Question: if you are the Chargers, do you go for two, or kick the extra point? The announcers were debating the issue, and I’ve seen more debate online, but there is actually a mathematically correct answer here.
The arguments were: (pro) if you go for two, and make it, then the game is over. If you don’t, the Broncos still have to drive the length of the field; not likely. (con) Kick the point, then the Broncos have to drive the length of the field – not likely – and then make a two-pointer, also not likely.
Three assumptions. (i) If the Chargers go for two and make it, then the Broncos have a negligible chance of winning. (ii) If the game goes to overtime, each team has a fifty-fifty chance of winning. (iii) The odds of making it when you go for two are less than fifty-fifty.
It’s easy enough to write out some equations that reveal the truth, but there’s an elegant way to argue this. In either case, in order for the Broncos to tie the game, they have to both drive the field and score a touchdown, and there is a two-point conversion that has to go the right way. The only difference is that if the Chargers go for two, they are the ones that have to make the conversion, whereas if they kick the point, the Broncos are the ones to go for two. Since the chances of making the two-point conversion are less than fifty-fifty, the best choice for the Chargers is to force the Broncos to have to make the conversion, i.e. the Chargers should kick the point.
Not sure that’s an especially clear explanation. Here are the formulas. Say the probability of the Broncos driving the field for a touchdown is TD, and the probability of a team converting the two-pointer is C. Then if the Chargers go for two, the Broncos win with probability
(1 – C) * TD [Chargers miss the two-pointer and the Broncos score]
and if the Chargers kick it, the Broncos win with probability
TD * C [ the Broncos score and make a two-pointer]
Since C < ½, the latter number is going to be smaller, which is what the Chargers want. So they made the right choice – go figure.
posted by Jim at
9:58 PM
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