Probability of Dependent Events: NBA Draft Edition
That’s what’s going down. Probability and the NBA Draft, y’all.
Many people have, at one point, dreamed about being a star athlete who is about to be drafted to the professional league of their choosing. While the dream might look different for each person, the main constant in this dream is being drafted. It doesn’t matter if that dream goes into the process of working out and interviewing for different teams, having a (pandemic-appropriate) party with your immediate family, or whatever cut-scenes NBA 2K has for that year’s MyCareer story, the dream is based around being drafted. The latter is usually my closest reality to actually getting drafted, but that’s not important, now, is it?
Let’s try an exercise: For a moment, try to imagine the anxiousness of waiting for that moment where you’re drafted. Then, imagine having to contain that emotion for upwards of at least four hours. Is it hard for you to imagine how long four hours truly is? Imagine feeling this way for at least 11 episodes of SpongeBob SquarePants. Sounds like a lot of hoopla.
If I was a draftee, I would ask tons of probability questions during this long anxious wait to ease my nerves. I think the obvious first question would be.
What is the probability I am selected as the number 1 pick in the draft?
If you haven’t read my previous articles on probability, I suggest you start there, as I cover the basics of probability and independent events. For those that did their homework (Is this an old bit yet? No? Okay, cool.), we’ll continue!
Since all drafts are different, I’m going to answer the questions in this article using my favorite league’s draft: The NBA Draft. In the 2017 NBA Draft, 182 players signed up, but out of those 182, only 60 players were actually drafted. Since all players are eligible to be drafted, calculating the probability of being drafted first is the same as calculating the probability of an independent event. For the sake of this article, let’s assume that all players have an equal probability of being drafted №1. The probability of being the №1 pick is: (1/182) =0 .00549 or slightly above half of 1%.
It is unlikely to be the number one pick in the NBA draft but I have to imagine every player hopes it’ll be them and not being it, probably adds to the anxiousness of the wait to be drafted.
What is the probability of being selected either №2 or №3 in the NBA Draft?
The Top 3 draft picks in the 2020 NBA Draft have contracts that are each worth at least $30 million. So, you could say they’re doing okay financially. In the 2020 NBA Draft, picks №4 — №30 all had starting salaries smaller than $30 million, so I’d assume a lot of players would be wondering what their probability of being one of the next two (№2 & №3) picks is if they’re not going №1.
In all drafts, a player can’t be drafted twice and is removed from the draft pool after being drafted. This means the event of being selected at any pick in the draft after the first pick is a dependent event. That’s because the probability of getting selected at any pick of the draft changes depending on the outcome of another event, or in this case, the previous pick.
To better understand this, let’s look back at the probability of being selected first overall. The probability was 1/182 because there is only one pick and in this hypothetical scenario, each of the 182 eligible players has an equal opportunity to be chosen №1. This is no longer the case for the second pick, as the first pick in the draft can no longer be drafted. To calculate the probability of being the №2 pick in the NBA Draft, I must solve for (1/181), which is equal to 0.00552, or even more slightly above half of 1%. Though it is still highly unlikely to be the second pick in the draft, it is slightly more likely than being the first.
In fact, since 60 players are selected in the NBA Draft each year, the best probability of being selected to play in the NBA is during the last pick. That’s because the eligible pool of players will be at its lowest. In the NBA Draft, the last player picked has a probability of [1/(182–59)] or a probability of 0.008 of being selected. Kinda sad when a 0.8% chance is the highest probability, wouldn’t you say?
Now that the probability of being the second pick in the draft is known, the probability of being the third is next, which is 1/180 or 0.0555. For those that aren’t great with math, that’s even higher than the 0.00552 from earlier. That means it’s even MORE slightly above half of 1%. It has already been established that since a player can’t be drafted twice, the calculation of the probability must respect the pick’s dependence on previous events. However, this also means that being selected at any pick other than the first pick is a mutually exclusive event. This is going to be on the test, so pay attention! Mutually exclusive events are events that can’t take place at the same time. An example could be being drafted №2 and №3 in the same draft. Meaning, in the 2019 NBA Draft, Zion Williamson could not be selected №1 and №2, obviously.
Or probability of mutually exclusive events: P(A or B) = P(A) + P(B)
To calculate the probability of mutually exclusive events, you simply have to add their probabilities together since they can never both occur. That means the probability of being the first or second pick in the NBA Draft is the probability of being the №2 pick (0.00552) plus the probability of being the №3 pick (0.00555). So, in the 2017 NBA Draft, eligible players had a 0.01 probability, or just over a 1% chance of being selected either second or third and earning an extensive contract. Don’t tell the kids that though. Let’s keep that one a secret for now.
Though there is a lot of money on the line during the draft, I guarantee there are plenty of eligible players that are nervous for other fellow players going through the process. They may have teammates or close friends who have also entered the draft. Many sports fans have seen this recently with the college superstars of Zion Williamson and R.J Barrett who were both selected №1 and №3 respectively. I’m sure before the draft, any young athletes in this position would ask all sorts of hypotheticals questions with their fellow athletes. One of those questions could be something along the lines of, oh, I don’t know:
What is the probability either my college teammate or I get drafted №1 or №2 of the NBA Draft?
This question may seem very similar to the question above, but it has one minor difference. Since both players can be drafted as the №1 and 2 picks, they are non-mutually exclusive events. The question also asks what the probability of either of the players getting drafted №1 or №2 in the NBA Draft. Due to that phrasing, only the case where one player is drafted in the Top 2 will be considered, not both of them in the Top 2.
Or probability of non-mutually exclusive events: P(A or B) = P(A) + P(B) − P(A and B)
To figure out this probability, I simply take the probability of all the events occurring minus the probability of both happening. Since we now have two players that could be selected as the first overall pick, the probability becomes 2/182, and the probability of the second overall pick becomes 2/181. Now, I must consider the probability of both events occurring, or [(2/182) * (2/181)] or 0.0001. That means the probability of one of the players getting drafted №1 or №2 is 0.021, or slightly over a 2% chance that one of the two players will be selected in the Top 2.
The knowledge learned in this article about dependent probabilities can help any anxious athlete handle the draft. They simply have to remember that as more picks occur their odds to be next increase.
