I’ve often thought about the football situation FSK describes in NFL Fnords. A football team scores a touchdown with no time remaining, an extra point will tie the game and send it to overtime, but a successful two-point conversion will win it. I always say “Go for the two point conversion!”
There is rarely a better opportunity to win a football game than from the 2-yard line. Going for two points gives you an immediate opportunity to win the game, your chances are in the neighborhood of 40%.
If you go for the tie, you’ve only got about a 95% chance of making the kick, and a 50% chance of receiving the kickoff. You only have a 45% chance that you’ll even have the first opportunity to score in overtime, there is a 5% chance that you’ll lose the game because you missed the kick, and a 50% chance that the other team will get the ball first in OT. (Note: winning in OT does not appear to be strongly correlated with the result of the OT coin flip, I think it’s about 50/50)
That notwithstanding, your odds of winning in OT are dependent upon either:
- putting together a drive long enough to kick a field goal, where your kicker has an 80% chance of success.
- putting together a drive long enough to score a touchdown.
- preventing the other team from doing either (1) or (2) and then doing (1) or (2).
Both of the first two scenarios depend on a series of successful plays, no turnovers, no penalties, etc. The game-winning OT touchdown drive, assuming that your RB doesn’t break one for 20+ yards, will probably come down to a play from scrimmage at the 2-yard line, in which case you’re right back where you started. This scenario is objectively less likely a priori to win the game than just going for the two-point conversion in the first place. The field-goal scenario may be your best chance, but remember that you’ve got only a 50% chance to receive the kick in OT. If you lose the coin-toss, you’ve got to stop the other team from doing either (1) or (2) and then successfully score a touchdown or a field goal of your own.