Introduction
It seems to me that this fledgling season has brought lot’s of comments on how Ross and Clarkson should drive more instead of taking 3 pointers. I think some of the reasons for these comments (although I shouldn’t speak for anyone else) are:
1. Clarkson and Ross are really good at attacking the basket
2. The new rules are resulting in more defensive fouls
3. Clarkson and Ross sometimes take poor 3 pointers, i.e. not in rhythm, defended, NBA range
It appears to me that 3 pointers can be overvalued because the expected value of a 3 pointer is seemingly larger than a 2 pointer. The weighted 3 pointer percentage for our Big3 is 0.358 (explained in detail later). Thus the expected value of a 3 pointer is 0.358*3 pts.=1.075. The weighted 2 pointer percentage for our Big3 is 0.482, thus the expected value of a 2 pointer is 0.482*2=0.963. So 1.075 > 0.963, right? However this is obviously incorrect because not all the possible outcomes are taken into account, e.g. being fouled. If all possible outcomes were examined, could the true expected value of a drive to the basket be actually higher than a 3 point shot?
Methodology
I will attempt a simple probabilistic calculation on the expected value of a drive to the basket versus a 3 pointer. I will focus on our Big3 (Clarkson, Brown and Ross), since they are quite capable at driving to the hoop, and they take the most 3s. I will use our season stats, although I will need to make some assumptions about unavailable stats.
Enumeration of outcomes
The figure below is a crude enumeration of possible outcomes after a 3 pointer or a drive to the basket. It’s crude because I did not include offensive foul or turnovers. Note that a blocked shot is included in a miss. I will attempt to calculate some probabilities associated with each branch of the probability tree along with the value associated with the outcome.
3 Pointer (or Drive)
________/__________\________
___Not fouled______Fouled____
___/______\________/_____\___
Make___ Miss___Make___Miss_
I will assume that the probability of a foul while attempting a 3 pointer is fairly small. Although we were treated to this not too long ago, when I believe it was Clark that was fouled while attempting a 3, and then seconds later he fouled someone also attempting a 3. I will assume that the probability of a 4 point play is even smaller. This is not too restrictive of an assumption, as I can always include this outcome later.
I will use the season stats:
Clarkson 
Brown 
Ross 
Weighted Big 3 

3PT FGA 
29 
60 
45 

3PT Pct 
0.276 
0.417 
0.333 
0.358276 
2PT FGA 
132 
106 
90 

2PT Pct 
0.515 
0.491 
0.422 
0.481726 
FTA 
43 
60 
55 

FT Pct 
0.884 
0.767 
0.709 
0.778652 
The last column represents the weighted percentages (percentages weighed by the number of attempts) for the Big3.
Unfortunately, I cannot find stats for the percentage of drives that lead to a foul, or the percentages of "and one" if fouled. Therefore I need to make a guess for the former.
Let’s assume that 2/3 of the drives there is no foul, i.e. the other 1/3 result in a foul.
Let’s also assume that of the likelihood of an "and one" is very small, i.e. negligible.
Now it’s time to enumerate the probabilities for the branches and calculate an expected value for a drive.
expected value of a nonfoul drive is 0.482*2 pts = 0.963
expected value of a foul is 0.779*2 attempts = 1.58
Then the expected value of a drive is 2/3*0.963 + 1/3*1.58 = 1.161
Conclusions
If my assumptions are somewhat on target, then this would mean that the expected value of a drive, 1.16, is larger than the expected value of a 3 point shot, 1.07. So ATTACK guys!
I assumed that the 2PT Pct is the same as a drive Pct which is obviously wrong. The value should be higher, but I don’t know by how much. It’s not as high of a percentage as a layup, but 0.482 (2PT Pct) just feels a bit low.
I understand that I had to guess at some values. But I don’t think they were completely foolish. For example, I tried to approximate the percentage of fouls from drives by using this ratio:
(FTA/2) / (FTA/2 + 2PT FGA). I divided FTA by 2 since a foul could result in 2 attempts (depending if it was in the act of shooting or not, otherwise a bump during the drive is a "one + one"). This gives the value of 0.194 using our season to date stats for our Big3. But I feel this value is an underestimate, since not all 2PT FGAs are drives. That’s why I picked 1/3. And that could even be a conservative ratio given the number of fouls being called this season.
In addition to expected values, the variability also matters. To use an economic analogy, it’s not just the rate of return that matters but also the risk. So one could consider that there is greater variance in the outcome of 3 pointers compared to a drive. Thus there is a lower risk associated with attempting a drive. I think when the game is tight, risk or variability comes into play more.
The whole strategy with fouls is a very complicated issue. Getting fouled by the opposing bigs is usually a good thing, especially if they have a highly skilled one (e.g. what we did to Blake Griffin and OU). But our player can also get an offensive foul if out of control. The number of fouls also determines the onset of the one+one and the double bonus, thus it could be significant late in games. I am unsure how to quantity the overall foul strategy into a per drive basis.
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