Quick Math: When it makes sense to go for the win or go for the tie

In week seven of the NFL season, the Tennessee Titans and Los Angeles Chargers game was coming down to the wire. With Chargers winning 20-13, the Titan's QB Marcus Mariota scored a touchdown with 34 seconds left in the game to make it 20-19. Instead of kicking an extra point to send the game into overtime, the Titans made the move to go for the win, trying to convert a two point conversion. They failed on the attempt, and lost the game.

Of course, every NFL talkshow exploded. Analysts spent the next 30 minutes roasting the Titans coaching staff on a stupid decision. It is hard to determine if the Titans successfully converted the two point conversion if the analysts would say "I love the aggressiveness" and "Great call", but most of them hovered around the theory "I don't care what the stats nerds say, that wasn't the correct call".

With the large surge of analytics and probability theory in sports, a lot of people don't understand how that logic is implemented, especially the TV personalities. This doesn't just pertain to the NFL though. In a NBA game, if you're down two with four seconds left on the clock, should you write up a play to have a high percentage shot to send the game into overtime, or should you shoot a lower percentage three point shot for the win? These situations usually lean towards taking the big boy shot and going for the win. But why?

Let's use the above NBA example first. Down two, 10 seconds left, you have the ball. The first thing we need to do is assess what are the probabilities of the following three outcomes:

1. Probability of making a two pointer
2. Probability of making a three pointer
3. Probability of winning in overtime

My favorite basketball team is the Denver Nuggets, and looking at their 2017-18 shooting statistics they made 52.5% of their two point attempts and 37% of their three point attempts. In that season, they were also 3-4 in overtime games, which is a winning percentage of 42.9%. Now having those three values, should they shoot a three for the win or try and force the game into overtime?

Trying to force the game into overtime requires you to make the two pointer, and also win in overtime. Since those are independent events, we multiply the events together to get a probability of 22.5% ( .525 * .429 = .225). Now rethink the problem: you have a 22.5% chance of making the two pointer and winning the game in overtime. You have a 37% chance of making a three pointer. Which one seems more reasonable? Obviously, shoot the three.

Now let's shift the focus back to the Titans vs. Chargers game. Instead of looking at individual stats of the Titans, we can take a bigger sample size and look at statistics of these categories across the league.

The "easy" option would first be kicking the extra point, but the simple 1 point to send the game into overtime isn't as easy as it looks anymore. Kickers are having a historically awful year, converting successfully 94.8% of the time (used to hover around 99%, which is why they moved it back from the two yard line to the 15 yard line back in 2015).

The two point conversion seems like a risky move compared to kicking a field goal, but through the first seven weeks, teams are 34-59 on going for two, which is a success probability of 57.6%. Not a horrible alternative since the odds are still in the favor of the offense.

How would you calculate winning in OT in the NFL though? Wouldn't it be 50/50 since there has to be a winner and a loser? Not exactly...since you can tie! There have been 10 overtime games in the NFL this year, and two have ended in a tie. So four wins, four losses, and two ties gives a winning percentage of 40% in overtime.

Doing the same calculations as before, the probability of converting a extra point and then winning in overtime is 37.9% (.948 * .400 = .379). Comparing that sequence of events to a two point conversion, you are 20% more likely to convert a two point attempt successfully than you are to winning a game in overtime.

I can understand the pressure of being in the all or nothing situation that the Titans were in, and good for them for actually doing the mathematically smart thing! If a two point conversion was about 5% more than winning in overtime, then I can understand the NFL postgame analysts' point of view. If you showed that the the odds of you winning in either situation were both below 50%, I would understand. These NFL analysts aren't supposed to say anything outrageous that makes them look stupid, and I can understand.

But the outrageous thing they're claiming is an extra point to go into overtime IS the smart play! It would make more sense if they said go for it! If a team has four times in their 16 game season where they are put in the predicament "Do we go for the win or do we go for the tie", mathematically the number of wins to expect with going for the tie is 1.52 wins out of the four games; however, if you go for it, the expected number of wins increases to 2.31. The only situation that kicking a field goal makes more sense is if you have at least a 60.8% chance of winning in overtime, and you have to know that.

So the next time you're sitting around with your friends watching the game and this situation comes up, don't sound like an idiot. "Go for two" should be the only words out of your mouth.