Are Free Throws Independent? Should Jake Be Working at The Ringer? An Investigation.
It began with a simple tweet:
This tweet came on the heels of Damian “I Don’t Chase Rings” Lillard missing two critical free throws against the LA Clippers while down one with only 15 seconds to go.
Although most of the post-game commentary focused on the Paul George-Pat Bev-Dame Hate Triangle, as a serious student of the game, I disengaged from such frivolity and instead spent my efforts on the hard science of the STAT ( not Amare Stoudemire).
One of the top comments on Zach’s tweet questioned the basis of this 1.2% calculation (another top commented astutely pointed out that 1.2% is a probability and not odds, but I digress), wondering if the probability of Dame missing his first free throw and his second free throw were in fact independent events.
Why does this matter? Well, we want to make sure that Dame’s first free throws and second free throws aren’t falling into a co-dependent relationship, always crashing each other’s friend hangouts, constantly texting, and of course, not dependent on the outcome of one to determine the outcome of the other.
In more real terms, events are independent if the outcome of one event does not affect the outcome of the other event. In a stats class, a coin flip is the classic example — the probability of getting a heads on your first flip is 50%. No matter what happens, the second flip does not depend on the first flip at all — the chance of getting a heads on the second flip is still 50%.
So if we want to know the probability of A occurring AND B occurring, it is just the probability of A times the probability of B, or in more formulaic terms:
P(A and B) = P(A) * P(B)
What’s the opposite of this? Again, using some classic stats examples, it would be the chance of drawing two hearts in a row from a deck of 52 cards. The probability of drawing a heart on the first card is simply 13/52, or 25%. However, the probability of drawing a second heart actually changes because of we drew a first hear t— there is one less card in the denominator, and one less heart in the numerator. So the probability of both happening is now the probability of A times the probability of B given A:
P(A and B) = P(A) * P(B|A)
Now, probability questions are more interesting when you’re not just flipping coins (a classic quarantine activity) — so going back to the question of, “what was the probability of Dame missing two free throws,” in order to answer that, we need to know if we should be using that first formula or the second (noting that they are technically the same, just in the first formula, the probability of B given A is just the probability of B).
And it’s not clear which it is — I could imagine that NBA players are so automatic, that missing one free throw has no impact on their other free throw (and the general notion of hot-hand theory is one that has been explored already), but as the 2008 Westwood Freshman Team free-throw champion, I also personally believe that hitting your first free throw gets you in a better rhythm. And most importantly, was Zach Kram right or wrong?!?
To get answer, we have to go to the spreadsheets! To determine the relationship between free throws, I simply looked at data from the the 2017–2020 season, and how players performed on their second attempt overall, versus how they performed on their second attempt given that they missed their first attempt.
And the survey said: players shot 78% on all of their second free throws, but only 74% on their second free throws, given they missed their first! Putting that difference to the test — chi-squared test that is — confirmed that the difference there is significant to call Zach Kram and a FREAKIN —
Wait a minute — he tweeted about DAME, not the whole NBA, some Twitter misanthrope is sure to point out. And that theoretical misanthrope has a point — looking at Dame’s percentages, he shot 91% on all his second free throws, and actually shot 92% on his second free throw following a missed free throw.
So what gives? Well, when I took a look at player-level free throw independence, it actually appeared that out of 101 players with over 250 second free throws taken during this time span, only 8 actually showed a dependent relationship between their second and first free throw attempts. My guess is that instead, what is driving the discrepancy between the 74% and 78% is that when we look at second free throws given missed first free throws, we are actually biasing the sample towards players who are worse free throw shooters overall (they missed their first free throw for crying out loud!).
So where does that leave us? It leaves ME having written nearly 900 words about a tweet sent out a month ago and nothing to show for it, but it leaves US with the comforting knowledge that first and second free throws are living healthy, independent lifestyles.