Probability of Independent Events with Steph Curry — Part 2
Continuing the intro to probability with the Golden State Warriors.
Last week, I posted an article quantifying how great of a shooter Steph Curry is, and how insanely low his probability of making 77 three-point shots in a row truly is. Spoiler if you haven’t read the first article: It’s incredibly unlikely. Like super, super unlikely. If you haven’t read last week’s article, you’re potentially not going to know what I’m referring to. However, I’ve got you covered. Here’s where you can find that article: Probability of Independent Events with Steph Curry. Those that have read the first one, let’s continue with our friend, Mr. Steph Curry — shall we?
The first article, as you know since you read it, focused a lot on Steph Curry. Since reflecting on the fact that I didn’t include Klay last time, I’ve devoted this article to right my wrongs. I will use both of the Splash Brothers, Steph, AND Klay, to teach anyone how to easily calculate the probability for multiple different independent events or expected outcomes. Are you ready for this? Let’s go.
Last time, the lesson was: How to calculate the probability of the same independent event having the same outcome for a given number of times in a row? This time, the lesson is about calculating the probability of different independent events having the same outcome. This article will also cover how to calculate probability using the same independent event but expecting different outcomes. Followin’ me?
The Splash Brothers are always breaking each other's franchise records. In January, Klay Thompson posted this hilarious video when Steph Curry beat his single-game scoring record by notching 62 points against the Portland Trail Blazers. With the light-heartiness it appears these competitors hold towards their records I wonder if they would ever consider breaking one together. Possibly even attempting to beat the amazing shooting feat we discussed in the previous article.
What is the probability of Steph Curry and Klay Thompson making 78 three-point shots in a row by switching the shooter each shot?
Remember, independent events are not impacted by the outcomes of any other events. In other, more simple words, Event A’s outcome (Steph Curry making a three-point shot) will not impact the outcome of Event B (Klay Thompson making a three-point shot) if they are independent events.
In this question, Steph would shoot first, then Klay, then Steph again, and so on until they had shot 39 three-point shots apiece. If it’s assumed the Splash Brothers attempted this incredible act in practice conditions, then I’ll use a shooting percentage that was recorded in near-practice conditions. In the previous article, I used a video of Curry making 93 out of 100 three-point shots during practice as his shooting percentage. Unfortunately, the best data sample I found for Klay Thompson in near-practice conditions is from the 2015 NBA Three-Point Contest that Steph Curry also competed in. To be fair, I’ll use both players’ best round in the contest to calculate their three-point shooting percentage.
In Steph’s best round, he went 19 for 25 (76%), while Klay was slightly better at 21 out of 25 (84%). To convert percentages to probability, you simply move the decimal over two places or divide by 100; this means the probability of Klay making a three-point shot during this hypothetical challenge is 0.84, and Currys’s is 0.76. Since it’s been established that independent events don’t have an impact on one another, it’s known that Steph Curry’s attempt is not going to affect Klay’s. That means the probability of the Splash Brothers making their first three-point attempts would simply be the multiplication of their probability of making the shot. Let’s get … mathematic.
Probability of Independent Events = Probability of ‘Event A’ success * Probability of ‘Event B’ Success
Now, that’s just the base formula. Let’s put Steph’s and Klay’s percentages in there and get to the bottom of this! Here we go: [(0.76) * (0.84)] = 0.6384, or a little over a 50/50 chance (63.8% to be more specific) that they both make their first attempts. Seems like a pretty likely thought. Having watched The Splash Brothers many a time, so seeing that number isn’t crazy to me. What about you? Let me know in the comments BELOW! Like, comment, subscribe. Alright, back to that math stuff!
There’s a problem, everyone. It’s still unknown what their probability would be when attempting 78 total shots (39 per Splash Bro). Those that come from a coding background (It’s me; I’m “those”) could solve this with, what they call in the biz, a simple for loop.
The code above would provide you the correct answer, but it isn’t the most efficient since the computer has to do the loop one fewer times than the number of attempts. With a small sample size like this, it’s not too much on the computer, but once the number of attempts starts increasing, the speed of the response time slows down significantly. Simple Loops also don’t show a good understanding of probability. It’s like copying someone’s calculations on a calculator, getting the same answers, and saying you get that lesson. No ‘ya don’t, fool!
It’s been stated that both of the events (Event A & Event B) are independent, which means they can’t and don’t impact the others’ outcome. Since neither shot impacts the other shot, it doesn’t matter who shoots first, third, ninth, twentieth, or sixty-fourth. The equation is either [(0.84) * (0.76)] or [(0.76) * (0.84)]; it doesn’t matter because that’s how multiplication works!
Probability of Successful Outcomes of Switching X and Y Events for some attempts in a row = (X’s Probability of Success * Y’s Probability of Success)^(Attempts / 2)
The probability of Steph and Klay accomplishing the hypothetical question is [(0.76) * (0.84)]³⁹ which is 2.503930414054546e-08 (aka two quadrillion five hundred three trillion nine hundred thirty billion four hundred fourteen million fifty-four thousand five hundred forty-six hundred-sextillionths). In English, The Splash Brothers would have to attempt this at least 39,937,212 times to expect one success. Of course, there’s a reason the game isn’t played on paper; anything can happen! As seen in the last article, in practice conditions, Steph Curry shot 93% from “behind the arch”. So, in theory, calculating their Three-Point Contest percentage lowers their chances of accomplishing this challenge. It’s seen in the previous article that any small difference between the success or failure of an event will cause the probability of that same outcome to happen X amount of times in a row to decrease exponentially.
The above covers when one wants to calculate the probability of two different independent events but what about when one wants to calculate the probability of the same independent event providing different outcomes.
What is the probability of Steph making and missing 39 three point shots in 78 attempts?
Since Curry’s two possible outcomes for a shot are only a made shot or a missed shot, and probabilities add up to 1, there’s an easy solution to find out this answer. The way to calculate Steph’s probability of missing is (1 — Probability of Make) or (1–0.76), which equals 0.24, or 24%. It’s already been established that making a shot is independent of each other, so now I can simply reuse the method above to answer this question. The probability of Steph making 39 three-point shots and missing 39 three-point attempts is [(0.76) *(0.24)]³⁹.= (0.1824)³⁹ = 1.5134515612278483e-29 or 1-in-6,607,413,316,807,508,994,8914,483,200 (sixty-six octillion seventy-four septillion one hundred thirty-three sextillion one hundred sixty-eight quintillion seventy-five quadrillion eighty-nine trillion nine hundred forty-eight billion nine hundred fourteen million four hundred eighty-three thousand two hundred) attempts. Sixty-six … Octillion.
It may seem, based on intuition, that Steph would be more likely to both miss and make 39 three-point shots than for Steph and Klay to make 78 in a row, but that is just how incredible of shooters they are. Like, yeah the Splash Brothers nickname is kinda cheesy, but it is spot-on. They’re quite literally the brothers of the splash. Not only is that my opinion alone, but there are literal stats, both in this article and elsewhere, that proves how incredible they are. Since Klay’s probability of making a shot is a lot higher than Curry’s chances of missing, the difference in the probabilities grew exponentially as the attempts went on, which caused the needed attempts to complete said challenges to be so drastically different.
This completes the lesson on the probability of independent events. Next time, we’ll explore the probability of dependent events. Get hyped. It’s about to be crazy. I’m excited just thinking about it!
