# Q1: What is the probability that the Lakers miss the playoffs?

Inspired by Ben Falk’s Predict challenge on his website, Cleaning the Glass, (a really cool website if you’re an NBA fan) and filled with intrigue about year 1 of the LeBron led Lakers experiment, I decided to tackle this question.

So, how should we go about answering it?

Well, I could build a complicated plus-minus based model. But I’m lazy. Others have already built really good models. I wanted to try a simpler approach.

Let’s start by looking at the betting markets. According to Vegas Insider, on October 10th the Lakers were -400 to make the playoffs and +300 to make the playoffs. What this means, for non-bettors, is that if you want to bet on them to make the playoffs you have to risk $400 for the possibility of winning $100. Likewise, if you bet on them to miss the playoffs, you risk $100 to possibly win $300.

What this means is that you would have to believe they have a greater than 80% chance to make the playoffs to bet on the former or believe they have less than a 75% chance to bet the latter. So maybe Vegas thinks the true probability of LA making the playoffs is somewhere in the 75% to 80% range?

To establish my own probability, I looked at how each team performed relative to their pre-season season win total over/under line, since the 2012-13 regular season. As it turns out, if you run a linear regression to predict regular season wins based on the over/under line you get the following equation:

Predicted Wins = 4.22 + 0.89 * Season Win Total Over/Under Line

This means that teams win, on average, about what the over/under line is set at, with a little regression towards the mean. The plot below shows how each team in the dataset fared relative to the prediction from this equation.

The really important part of this regression is actually not the equation but something called the residual standard error. The residual standard error basically tells us how much variability there is in the actual wins of a team around our predicted value. The residual standard error of our regression was almost exactly 8 wins, which means that our 95% confidence interval for predicted wins is the prediction from the equation *plus or minus 16 wins. *

That’s a huge amount of variability, which really surprised me! This mean that if we predict a team will win 41 games then we can only be 95% confident that they will win somewhere between 25 and 57. This agrees with our data. Since the 2012-13 regular season, 6 of the 179 teams have won or lost at least 16 more games than we would expect based on the regression equation. 6 divided by 179 is about 3.4%, not far off from the 5% we would expect.

So what does all this math have to do with the Lakers and the playoffs? Here’s what: to estimate LA’s playoff odds, I started by taking each team’s season win total over/under line, from Westgate. Next I computed expected wins from my regression equation. Then, I estimated that each team’s true win total is normally distributed about this expectation with standard deviation 8 wins (the residual standard error from before). I used this estimate to run 100,000 simulations and recorded how many times each team made the playoffs. Here are the results for the West:

* According to Westgate, on October 10th.

** Found using the equation from the linear regression: Expected Wins = 4.22 + 0.89 * Season Win Total O/U Line

*** Computed via 100,000 simulations, assuming win total is normally distributed around expected wins with standard deviation of 8 wins.

If you want to see the derived playoff probabilities for the Eastern Conference teams too, you can go to this google sheet.

So this method tells us that the Lakers have a 73% chance to make the playoffs and a 27% chance to miss the playoffs. That’s about what I would expect. You could argue that this year’s Lakers squad has even more variability in win total than your average NBA team, which would increase their chances of winning many more games than we would expect but also increase their chances of falling short and missing the playoffs. So maybe their chances of missing the playoffs are a little more than 27%.

On the other hand, you could argue that the real possibility of Jimmy Butler being traded to the Eastern Conference means my method is undervaluing their playoff odds. I think Westgate is already pricing the Butler trade possibility into their lines, as the T-Wolves opened at a 44.5 and are now down to 41.5. Still, if you think the Timberwolves are really more of a 38 win over/under team without Butler (which is probably too low) and run the simulations than the Laker’s probability of missing the playoffs falls from 27% to 25%. It’s not a big difference. You could you could argue that the Timberwolves will be, on average, much worse than a 38 win team without Butler, but I’m not quite buying that.

*Answer: I estimate that the Lakers have about a 27% chance to miss the playoffs, about the same chance as flipping tails twice with a fair coin.*

# Q2: When is it safe to leave an NBA game at the start of the 4th quarter?

Say it’s a blowout at the start of the 4th quarter. You want leave the arena and go home, (or more likely turn off the TV) but you don’t want to wake up and find out that the trailing team ended up coming back and winning. How large a lead do you need to safely go home?

To answer this question, I used basketball-reference data from the past two regular seasons. I built a (logistic regression) model to predict which team will win the game based on the score at the start of the 4th quarter, which team is home, and the difference in ability between the two teams, in terms of points of net rating difference.

So, if the two teams are roughly equal in skill, then a 15 point home lead or 18 point away lead is needed to be 95% confident that we won’t miss a comeback. If one team is a bit better than the other, then you should add (or subtract) a few points from these numbers, depending on who is leading.

*Answer: You can be 95% confident that you won’t miss a comeback by the trailing team if the home team is leading by at least 15 points or the away team is leading by at least 18. If one of the teams is substantially better than the other, then you should adjust these thresholds by a couple points.*

# Q3: Say I’m a Kings fan. Will they win a championship in my lifetime?

Alright tortured Kings fans, here’s a question for you. I feel for you guys- I’m a fan of the Brooklyn Nets.

To give us somewhere to start, let’s say that you are an 18-year-old Kings fan. Not to be too morbid, but for 18-year-old males the Social Security Administration estimates that you will live, on average, another 59 years. If you are a female, then you live an extra 4.7 years on average.

Now, we know the Kings are almost certainly not going to win it all this year (Bovada has them at 500 to 1). But in 2050? That far in the future, we might estimate that they have as good a chance as any other team, that is 1/30 ~= 3.33%.

To account for this high likelihood of near-term futility but high uncertainty in the long term, this is what I decided to do: let’s give the Kings have a 1/500 chance to win the championship each year for the next 5 seasons, a 1/100 chance for the following 5 seasons, and then a 1/30 chance (i.e. league average) in each subsequent year after that. (I assume each season is an independent event). This gives your 18-year-old male Kings fan a 1 – ((499/500)^5 * (99/100)^5 * (29/30)^49) ~= 82% chance to witness a Kings championship in their lifetime. A female 18-year-old has about an 85% chance. Have hope Kings fans!

This is just a rough estimate that tries to account for two ideas: they are almost certainly not winning a championship in the next few years, but have as good a chance as any other team in seasons far in the future. You can of course disagree with my estimates. Maybe you instead want to say the Kings only return to ‘league average’ championship odds 20 years from now, rather than 10. Then you arrive at a 77% chance of a King’s finals victory in the lifetime of the 18-year-old male fan. That’s not too large a difference.

What about if the NBA expands and each individual team thus has lower championship odds? What about possible relocation? What about the chance the NBA isn’t even around in 30 years? If you want to factor these possibilities into your analysis then you are more dedicated than I am.

*Answer: Probably, especially if you are relatively young. An 18-year-old male Kings fan has about an 82% chance of seeing the Kings win a championship in their lifetime.*

### References:

Data used in this post was from basketball-reference.com.