Nylon Calculus 101: Expected Value and Shot Selection

The most visible on-court impact of basketball analytics is on shot locations. As numerical analysis has flourished, it has become broadly accepted that threes and layups are good shots while mid-range shots are not.[1. Everything else being equal.] In taking a more mathematical approach to studying the game, this concept was a bit of low-hanging fruit,.

At the core of the “avoid-long-twos” idea is a concept that is as omnipresent in basketball as it is in everyday life: Expected value. Expected value is, without using nerd-speak, the numerical summary of all possible values an event can take. In essence, it is a weighted average.

Suppose I offered you a chance to play a fantastic game. I have a fair coin[2. That is, one equally like to land on hands or tails.], and I offer to pay you $20 for heads, if you pay me $1 for tails. I flip the coin, it lands tails, I laugh and take your dollar. Was it a mistake for you to agree to play?

You might say yes, because hey, you’re out a dollar, how was playing the game smart? But the answer is no. Just because you lost once, doesn’t mean it was dumb to play. In fact, you should be willing to play not only that game, but also play it for a dramatically smaller payout ratio. As is, you’re making $9.5 every time the coin is flipped, since we expect the ratio of heads and tails (over the long run) to be almost exactly 1:1. For every $1 you lose on tails, you’re making $20 back on heads, two events which occur in equal amounts, giving your coin flip an expected value of $9.5.

If this sounds like a casino, it’s because it is. Entire industries (casino gambling, lotteries, insurance) are built off this idea. Sure, someone might come over to a roulette table, throw down a grand on one number, and walk away one spin later with $35,000. But there are (usually) 38 numbers on a roulette wheel, (1-36, 0, and 00), and thus, the fair payout for hitting one number is 38:1, not the 35:1 payout most casinos offer. Lucky hits like this happen every minute in a crowded casino, but the casino knows all about expected value, and is happy to take the expected difference between the fair price and their payout offering as profit.

The reason that folks who look at basketball numbers advise against taking mid-range shots is because of, you guessed it, expected value. Take a look at the table below, which uses 2014-2015 NBA league average data from basketball-reference.com.

Since the value for a missed shot attempt is simply zero points[3. We are intentionally ignoring the effects of offensive rebounding for the moment. It will be the subject of discussion later on in the syllabus, but a missed shot does have some positive value to the shooting team via the chance of offensive rebounding, which makes a miss a more valuable play than a turnover, which has an expectation of exactly zero.] , the expected value of a shot attempt is simply the product of the shooting percentage and the amount of points given for a made shot. For the league average player, the best shots, the shots with the highest expected value, are quite comfortably, shots within 0-3 feet from the hoop and three pointers. Though three pointers are made less than any other shot, the extra point provided for making one is goink[3. #Goink.] to more than make up for the fact that they are made slightly less often. In the long run, threes from a league average player are going to net you more points than the same amount of mid-range shots.

This isn’t to say mid-range shots should never be taken. If a league average player gets the ball at 16 feet with 2 seconds left on the shot clock, he should probably shoot it, as the expected value of his shot attempt almost certainly exceeds the expected value of everything thing else he can do in the remaining 2 seconds, and the expected point value of the possession overall is quickly converging to 0, the result of a shot clock violation. [5. The SportVU system allows for some examination of these decisions in real time.]

Players are also radically different with regards to how they shoot from different spots on the floor. Bismack Biyombo, who has shot 18.2% from three on his career, should probably never shoot from three, and his three pointer is only very slightly more preferable than his 27.0% career mid-range shot (0.546 expected points vs. 0.540 expected points). So each player’s shot selection prism will be unique to his skillset.

It is also important to note that there may be unquantified advantages and disadvantages beyond the amount of points scored for each shot type. In a sense, the numbers shown in the table above are only estimates of the expected value of each shot. Do mid-range shots provide a spacing benefit roughly equal to that theorized to be provided by three point shooting? Are shots taken from 10 feet away or further harder to contest than shots closer to the hoop and thus serve to more efficiently tire out the contesting defensive players? Is shooting from certain areas of the floor more valuable among certain types of players? These are questions the basketball analytics community is working to more definitively answer.

But as a general rule of thumb, the pervasive concept of expected value dictates that threes are vastly preferable to mid-range shots, and even preferable to shots taken from 3-10 feet away. The concept of expected value and this theory of shot selection are two of the most important ideas in basketball analytics.