Probability of Independent Events with Steph Curry
An intro to probability with the greatest shooter of all time
When thinking of the best videos to look up on YouTube, part of me wants to say that this video is not among the most accredited. How?! 77 made shots in a row and completing 93 out of 100 attempts. There are so many things out there in this world that happen less frequently than 93% of the time. Now, obviously, the sample size is only 100, but that is still wildly impressive. Give Steph the respect he deserves, you know?
In this article, I seek to explain how doing anything successfully 77 times in a row, without close-to-perfect odds, is nearly impossible. The way I’m going to do that is by somehow quantifying this video and mathematically proving how sensational it is. Now, I know math isn’t everybody’s cup of tea, but I’d say that more people like sports than math. Sorry, math people (I’m one of you). Now, once you finish reading this article, I hope that we can accomplish two goals, at least. Those two goals are: A.) learn some basic probability; B.) take a moment to appreciate the little moments in sports that make them amazing. Well, let’s get into the nitty-gritty, shall we?
To truly understand how incredible this video is, we need to calculate the probability of Steph Curry doing it again.
In 2015, Warriors’ point-guard Steph Curry hit 77 corner threes in a row during a team practice. His three-point shooting percentage for the 2015 season was 44%. Was that practice session a miracle, or is it probable to happen again?
Since Steph Curry making a shot does not affect his chances to make the next one, every shot he takes is considered an independent event. An independent event is simply an event that is not influenced by another event. For example, if I flipped a coin twice and the first attempt happened to land on heads, that first attempt would not influence the second one. I would still have a 50/50 chance of the coin landing on either side. Some basketball players would argue that making a shot increases your odds to make the next shot, even though they are independent events. This is a common fallacy known as the gambler’s fallacy.
This means that to calculate the probability of Steph making 77 shots in a row, one must simply take the probability of a true event (Steph Curry making the shot) to the power of attempts.
Probability of making X attempts in a row = Probability of making a three-pointer ^ Attempts
To complete the formula above, we need to know what the probability of Steph Curry making a shot from behind the three-point line is. Luckily, the question states his three-point percentage as 44%. This information can easily be translated into a probability, as a percentage is simply the amount of time you expect an event to be a success out of 100 attempts. That means, on average, it can be expected that for every three-pointer that Steph Curry shoots, he will make 44 shots and miss 66 shots.
Whereas a probability is simply a value from 0 to 1 that states how likely an event is to be a success out of one attempt. If we want to represent Steph Curry’s three-point percentage as a probability, we need to divide his percentage by 100 attempts. That’s because probability represents the likelihood of Steph Curry making one attempt. Followin’ me?
Probability of making X attempts in a row = (Three point percentage/ 100)^ Attempts
This means that our formula for calculating the probability that Steph Curry makes 77 shots in a row is (.44)⁷⁷, which comes out to 3.514439645386026e-28, or one expected success in 2,845,403,822,236,246,913,023,213,568 (2.8 octillion) attempts. Sounds easy enough, right? The odds of winning the lottery are 1 in 14,000,000 (14 million), meaning you are exponentially more likely to win the lottery than see Steph Curry make 77 threes in a row again. Read that again for me please. Yep, just read that again and then watch the video. Do you get why I am infatuated enough to do the math and write an article about this?
However, that is if it’s assumed that Steph Curry's uncontested shooting percentage is the same as his in-game shooting percentage. The two are rather different, I’d say. When Steph Curry, the three-point-making machine that he is, is statistically (almost) automatic. BUT, once you add the variables of a game, there’s no place left for that percentage to go but down. There’s fan noise (authentic or not), defenders, memory, tempo, court-tilt, etc. going on each game for all players, Steph Curry included. Since it’s not fair to assume that both shooting percentages (uncovered vs. in-game) are the same, the next goal is to figure out how to quantify a sample of Steph Curry shooting three-pointers in conditions similar to practice.
One of the closest samples of those conditions laid out previously is from the 2015 NBA Three-Point Contest. In that specific contest, Steph Curry made 19/25 three-pointers to ultimately crown him the champ that year. With that sample, his three-point-shooting percentage was 76%, or in probability terms, he had a 0.76 probability of making a shot attempt each time he shot the ball. Would we still be more likely to win the lottery than see Steph hit 77 threes in a row if his three-point percentage was 76%? The math to answer that question is (0.76)⁷⁷, which comes out to 0.6.647320299031722e-10 or 1-in-1,504,365,602 (1.5 billion) attempts. With a 1-in-1,000,000,000 (1 billion) chance, you are still more likely to win the lottery than see Steph Curry shoot as he does in the intro video.
As seen from the intro video, the conditions are certainly practice conditions. Since the video is out of 100 attempts, it’s super simple to convert to a percentage. Ready for this? Okay. In the video, Curry not only makes 77 three-pointers in a row, but he also makes 93 out of the 100 total attempts. If we use his shooting percentage of 93% in the video, that means to calculate his probability of making 77 in a row, we simply take (0.93)⁷⁷. The answer to that comes out to 0.0037427, or 1-in-268 tries. Not too shabby, eh? This may not seem like a lot of tries, but if we assumed that each attempt to make 77 in a row took the time of the video, which is 6 minutes, Steph would have to shoot for over 26 hours to recreate this feat at least once.
The Python code above can be used to quickly calculate the probabilities that we have found above and includes an optional Boolean flag to do the inverse calculation.
The inverse of a probability is the probability of an event not being successful. In our case, this would be Steph Curry not making an attempted three-point shot. This is the case since Steph can only have two outcomes of a shot attempt: it goes in, or it doesn’t. Since there are only two outcomes and probabilities are on a scale from 0-to-1, the probability of Steph missing a shot attempt is (1-[The probability Steph makes it]).
The Inverse Probability = 1 — Probability of making a three-point shot
It’s possible to use the algorithm from above to calculate what the probability was of the Houston Rockets missing an NBA-record 27 straight three-point attempts in 2018 versus Steph Curry and the Warriors’ no-miss streak. In 2018, the Rockets shot 36% from beyond the three-point line. That means the Rockets had a probability of 0.36 of making a three-point attempt and a probability of (1-.36), or 0.64, of missing the shot. The probability of them missing 27 in a row is (0.74)²⁷ or 5.846006549323615e-06; in English, that is twenty-nine million four hundred sixty-one thousand six hundred seven hundred-billionths. So, not very likely!
Not only did they miss a historic amount of three-pointers in a row (27), but it happened on one of the largest stages possible: Game 7 of the Western Conference Finals. This, statistically, should only happen to the Rockets 1-in-171,057 times they take 27 three-point shots in a game. So, on paper, we should never see that again. We’ll see if that actually holds. As a huge basketball fan, the fact that it happened in a deciding game to play for the NBA Finals is both amazing and tragic. Thankfully neither of the teams playing were my favorite team, otherwise, there would be a much different tone to this whole explanation.
As seen by the countless jaw-dropping moments throughout history, the sports world defies probability every day. Whether it’s Steph Curry making 77 three-point shots in a row, or the Rockets having a historic collapse in one of their biggest games ever, nothing is ever guaranteed in sports. We, as fans, get to continue to witness unimaginably unlikely events every season, and that is why we love the game. Anything can happen, even when the odds say, “Take a hike.” Hopefully, this helps you appreciate how talented these athletes are and how special it is to see them do what they do.
