(05-19-2021, 10:19 PM)Hogmelon Wrote: Yes, a 70% free throw shooter has a 9% [30% X 30%] chance of missing two out of two free throws, a 42% chance of making exactly one free throw [30% X 70% for miss-the-first/make-the-second, and 70% X 30% for make-the-first/miss-the-second], a 49% [70% X 70%] chance of making two out of two free throws, and a 91% [42% + 49%] chance of making at least one of two free throws.
So ...
Please explain what you mean by "the same logic probably holds for free throws."
In the OP, I wrote,
“Less obviously, it seems to me that in close games, it often matters more that you do score than it does how many points you get. So just like you ought to gradually move your retirement account into bonds and away from more volatile stocks as you near your personal end, doesn’t it make sense to think an offense ought to gravitate towards the higher percentage shots—and therefore, presumably, reduce exposure to volatility—as we move into and through the 4th quarter?”
What I mean by “same logic . . . free throws” is that, as the end of the game nears, FTs likely become even more valuable relative to 3-point attempts.
Let’s compare league average from 3 (36.7%), from 2 (53.0%) and from the free throw line (77.8%). On average, a 3PA has an expected value of 1.10 points (3 x .367), a 2PA 1.06, and a pair of FTs 1.56.
We could reasonably express the relative values of these shot attempts by saying, on average, the 3-point attempts are worth 104% of the value—and free throw pairs 147% of the value—of the 2-point attempts.
As you know, this is exactly why teams are taking fewer 2-point attempts these days and hoisting more and more 3s. And it’s why free throws (and the ability to get them) have always been a key factor in the outcome of games.
But let’s imagine that a hypothetical average 3 or average 2 or average pair of FTs is available on 4 consecutive possessions in a hypothetical game. Expected values for the shots stay the same, but because there are 4 possessions, we multiply by 4:
Expected value of 4 average 3PAs = 4.4
Expected value of 4 average 2PAs = 4.24
Expected value of 4 average pairs of FTs = 6.24
Now finally, let’s talk volatility. Here are the chances of going scoreless on those 4 consecutive possessions:
3PAs: 16.1% (63.3% miss rate x 63.3% x 63.3% x 63.3%)
2PAs: 4.9% (47% miss rate x . . .)
FTs: .00017% (22.2% miss rate x . . .)
Let’s state the same thing in slightly different terms. A team that shoots 3s on 4 consecutive possessions in the NBA will go scoreless on those 4 possessions about 1 out of 6 times. Shooting 2s instead will result in going scoreless on 4 consecutive possessions about 1 out of 20 times. Shooting FTs instead will result in going scoreless on 4 consecutive possessions about 1 out of 6,000 times.
This math seems to support what I was trying to say.