# Fiesta finish

Just like the Rose Bowl, the Fiesta Bowl had a finish I couldn’t resist commenting on (with some prodding from Adrian the Canadian). Stanford’s freshman kicker Jordan Williamson badly missed a field goal to win at the end of regulation, then missed again in OT to open the door for Oklahoma State. On their turn, the Cowboys got a quick strike to the Stanford half yard line (the refs initially called TD but it was overturned on review). Instead of going for the touchdown, Cowboys coach Mike Gundy elected to center the ball with a kneel and kick the field goal on second down.

Aptly named Quinn “Sharpshooter” Sharp booted the ball through for OK State, and the Cowboys went home happy. But should they have run the ball a couple more times first? The chance of a fumble is very low. The chance of Sharp missing the field goal is also very low. I expected this to be a semantic argument about 1 or 2 percentage points, but let’s do it anyway.

This was basically an extra point for Sharp. He was 74 for 75 on extra points this year (98.7%). I’ll round up to 99% for the “kick on 2nd down” strategy, since a botched snap would allow for another kick on 3rd down. So, the probability of a win by kneeling on 1st down and kicking on 2nd down is 99% (I’ll assume that there is no risk of losing the ball on a kneel down).

What if they decided to run for the touchdown on 1st and 2nd down and kick on 3rd down? Let’s say there’s probability f of losing a fumble on each running play and probability s of scoring a touchdown. So the probability of winning on 1st down is s. The probability of keeping the ball but not scoring on 1st down and winning on 2nd down is (1-f)(1-s)s. The probability of keeping the ball again but not scoring and winning with a field goal on 3rd down is (1-f)(1-s)(1-f)(1-s)0.99. So, the total probability of scoring (the sum of the probabilities on each down) is:

s+(1-f)(1-s)s + (1-f)^2 (1-s)^2 * 0.99

Let’s try s = 50% and f = 1%. This gives a probability of winning of almost exactly 99% again. To give the touchdown strategy a better chance, let’s try s = 70% and f = 0.25%. This gives a probability of winning of 99.8%. If they also ran on 3rd down and kicked the field goal on 4th down with these parameters, the probability would rise to 99.9%.

So, Oklahoma State might have slightly raised their chance of winning by running the ball, but their chance of winning was already so high (around 99%), that even quite optimistic parameters make little difference. However, what if the Cowboys were worried about the pressure on their kicker? After all, Williamson, a decent kicker, was visibly nervous and missed two pressure field goals for Stanford. Let’s say Sharp’s conversion percentage was only 90%.

In the “kick on fourth down” strategy with reasonable parameters (s = 60% and f = 0.5%), the probability of a win is still 99%. “Kick on third down” wins 98% of the time. Nerves are unlikely to affect running plays, so I think the Cowboys should have run the ball a couple times, on the off chance that their kicker was affected by nerves.