In the NFL it is my understanding that since the advent of the two-point conversion in 1994 the league-wide probability of making a two-point conversion has been roughly 50% and the probability of making the extra point has been close to 100%. For the below analysis I use the numbers 50% and 100% respectively. Please let me know (citing a source) if the former figure is significantly off.
The NFL game history seems to be packed to the brim with situations where coaches foolishly forgo the two-point conversion. Here’s an interesting scenario I just thought of while anguishing in the dentist chair:
4th quarter, very little time on the clock (let’s say right around 2 ½ minutes). Team A has just scored a touchdown to bring it within 8 points of Team B. Ceteris paribus, should Team A take the extra point or should it try for the two-point conversion? Well clearly, based upon the content of the preceding paragraph, I’m sure you'll realize I’m going to argue for the latter. Here’s why.
With the small amount of time remaining on the clock, it is reasonable to assume that Team A will need to do two things to win the game: the first is to prevent Team B from scoring again; and the second is to score another touchdown. (Furthermore, it is unreasonable to expect Team A (except in the most pathological of cases) to score more than one further time in the quarter.) Now if Team A is fails to accomplish these two tasks, it will lose the game. This is apparent whether it converts 0, 1, or 2 points after this last touchdown. However, if Team A IS able to accomplish these two tasks, the situation becomes more interesting.
Let’s say that Team A goes for the extra point which, as per assumption, it makes with certainty. After it scores its next TD, it is then down by 1 point. If it chooses to kick the extra point there, the game goes to OT which, given our assumption of all else being equal, it wins with probability 50% (neglecting ties). If instead of kicking the extra point after the second TD it goes for two, Team A will still win the game with 50% probability (i.e., if it converts (probability 50%) it wins, if it fails to convert (probability 50%) it loses). Hence, if Team A takes the extra point after its first TD and manages to accomplish the two tasks outlined above (preventing Team B from scoring again in the quarter AND scoring another TD), it will win the game with 50% probability.
OK, now let’s say that Team A goes for the two-point conversion which, as per assumption, it makes with 50% probability. If it converts (50% probability, all conditioned on its accomplishing the “two tasks”) it wins after taking the extra point following TD# 2, if it fails to convert (50% probability) then, down by 8, it tries for the two-point conversion after TD# 2. If it makes that (50% probability) the game goes to OT. In OT it wins with 50% probability (once again neglecting ties). If it fails to make the two-point conversion after TD# 2 (50% probability) the game is lost.
So in the case of Team A initially going for the two-point conversion, conditioned on its accomplishing the “two tasks”. The probability of a victory = prob of making first TPC + prob of not making first TPC * prob of making second TPC * prob of winning in OT = 50% + 50%*50%*50% = 62.5%.
So to sum up: down by 8 after having just scored a TD near the end of the fourth quarter, the DEFAULT action for coaches should be to go for the 2 points. A coach should only go for the extra point if team specific factors would so dictate (say perhaps if Team A is sufficiently stronger than Team B such that it’s probability of winning in OT is greater that 62.5%, or if the Team A short-yardage game and Team B’s red-zone defense are such that Team A’s chance of converting a TPC would be less than 41.42%).
QED
The NFL game history seems to be packed to the brim with situations where coaches foolishly forgo the two-point conversion. Here’s an interesting scenario I just thought of while anguishing in the dentist chair:
4th quarter, very little time on the clock (let’s say right around 2 ½ minutes). Team A has just scored a touchdown to bring it within 8 points of Team B. Ceteris paribus, should Team A take the extra point or should it try for the two-point conversion? Well clearly, based upon the content of the preceding paragraph, I’m sure you'll realize I’m going to argue for the latter. Here’s why.
With the small amount of time remaining on the clock, it is reasonable to assume that Team A will need to do two things to win the game: the first is to prevent Team B from scoring again; and the second is to score another touchdown. (Furthermore, it is unreasonable to expect Team A (except in the most pathological of cases) to score more than one further time in the quarter.) Now if Team A is fails to accomplish these two tasks, it will lose the game. This is apparent whether it converts 0, 1, or 2 points after this last touchdown. However, if Team A IS able to accomplish these two tasks, the situation becomes more interesting.
Let’s say that Team A goes for the extra point which, as per assumption, it makes with certainty. After it scores its next TD, it is then down by 1 point. If it chooses to kick the extra point there, the game goes to OT which, given our assumption of all else being equal, it wins with probability 50% (neglecting ties). If instead of kicking the extra point after the second TD it goes for two, Team A will still win the game with 50% probability (i.e., if it converts (probability 50%) it wins, if it fails to convert (probability 50%) it loses). Hence, if Team A takes the extra point after its first TD and manages to accomplish the two tasks outlined above (preventing Team B from scoring again in the quarter AND scoring another TD), it will win the game with 50% probability.
OK, now let’s say that Team A goes for the two-point conversion which, as per assumption, it makes with 50% probability. If it converts (50% probability, all conditioned on its accomplishing the “two tasks”) it wins after taking the extra point following TD# 2, if it fails to convert (50% probability) then, down by 8, it tries for the two-point conversion after TD# 2. If it makes that (50% probability) the game goes to OT. In OT it wins with 50% probability (once again neglecting ties). If it fails to make the two-point conversion after TD# 2 (50% probability) the game is lost.
So in the case of Team A initially going for the two-point conversion, conditioned on its accomplishing the “two tasks”. The probability of a victory = prob of making first TPC + prob of not making first TPC * prob of making second TPC * prob of winning in OT = 50% + 50%*50%*50% = 62.5%.
So to sum up: down by 8 after having just scored a TD near the end of the fourth quarter, the DEFAULT action for coaches should be to go for the 2 points. A coach should only go for the extra point if team specific factors would so dictate (say perhaps if Team A is sufficiently stronger than Team B such that it’s probability of winning in OT is greater that 62.5%, or if the Team A short-yardage game and Team B’s red-zone defense are such that Team A’s chance of converting a TPC would be less than 41.42%).
QED
