NBA Draft

Each year, the NBA teams recruit the best available talents – from colleges in the US or from overseas – through a process known as the draft. A draft order determines which team can choose first, second etc. Let’s divide the recruitment process into four stages.

Order teams

The 30 teams in the NBA are ordered in the reverse order of regular season record – the worst takes place 1, and the best gets place 30.

Draft Lottery

The top 14 of the previous list (remember: the worst 14 of the regular season) are eligible for the draft lottery. These 14 teams are the ones who miss out on the playoffs, i.e., 30 total – 16 playoffs = 14 remaining. The lottery is only to determine the top 4 picks. The 14 teams get probabilities of winning the lotteries based on where in the list they are.

Team Probability
1 (the last) 14.0%
214.0%
314.0%
412.5%
510.5%
6 9.0%
77.5%
86.0%
94.5%
103.0%
112.0%
121.5%
131.0%
140.5%

For example, here is the list for 2023 with the names and their respective probabilities.

Team WinProbability
Pistons.20714.0%
Rockets.26814.0%
Spurs.26814.0%
Hornets.32912.5%
Trailblazers.40210.5%
Magic.415 9.0%
Pacers.4276.8%
Wizards.4276.7%
Jazz.4514.5%
Mavs.4633.0%
Bulls.4881.8%
OKC.4881.7%
Raptors.5001.0%
Pelicans.5120.5%

You may notice a slight variation in the chances. That happens whenever two or more teams tie (same win%); the probabilities are added and divided equally (if the sum is odd, the division hands a slight advantage to one of the teams).

Picks 5 – 14 and 15-30

The 11 teams that missed out on the four lottery picks will get to pick players as per their order in the list. It means the No. 1 team on the list, if that misses all four lots, is guaranteed the No. 5 spot. The same goes for the 16 playoff teams – they get to choose 16 candidates (as No. 15-30). This concludes round 1.

Picks 31-60

The entire second round (No. 31-60), is determined by reverse order of regular season record.

The statistics of the lottery is in the next post

References

  1. How NBA Draft Lottery Probabilities Are Constructed: Squared Statistics
  2. NBA draft lottery: Wiki
  3. NBA Draft Lottery: Odds, history and how it works: NBA
  4. Tanking Won’t Die in the New NBA Draft Lottery System. It Will Only Evolve: The Ringer

NBA Draft Read More »

Republican Bayes

Let’s answer this question. In the Pew Research Center poll results published in 2010, 53% of Republicans, 14% of Democrats and 31% of Independents answered NO to the question, is there solid evidence that the earth is warming?
If a respondent answered no, what is the probability that she is a Republican? Note that on this survey on Oct 13-18, 2010, 25% of the participants were Republicans, 31% were Democrats, and 40% were Independent.

Let’s use the general formula of Bayes’ theorem here:

\\ P(j|N) = \frac{P(N|j)*P(j)}{\sum\limits_{i = 1}^{n} P(N|i)*P(i)}

Here, j represents Republican, and ‘i‘ represents a Republican, Democrat or Independent. So the required probability that a person is a Republican, given that she answered NO, is:

P(R|N) = \frac{P(N|R)*P(R)}{P(N|R)*P(R) + P(N|D)*P(D) + P(N|I)*P(I)} \\\\ \frac{0.53*0.25}{0.53*0.25 + 0.14*0.31 + 0.31*0.4} = 0.44

So, there is a 44% chance that the random person is a Republican: no better than flipping a coin!

Increasing Partisan Divide on Energy Policies: Pew Research

Republican Bayes Read More »

Three-parent baby

We have seen Mitochondrial DNA (mtDNA) as a valuable tracer to follow maternal ancestry. To take a step back: the majority of human DNAs reside inside the cell nucleus, and a few are inside another structure inside the cell, the mitochondrion. During reproduction (fusion of egg and sperm), nuclear DNA undergoes recombination with material from both parents participating, whereas mtDNA we possess entirely comes from the mother’s ovum. It happens due to the faster degradation of mitochondria from the sperm during fertilisation.

Leigh syndrome

Leigh syndrome is a fatal disorder, and its genes reside in the DNA of the mitochondria. If the mother has the disease, it’s sure to reach the offspring, jeopardising its health. In 2016 John Zhang’s team at the New Hope Fertility Center in New York City found a solution. They ‘swapped’ the mitochondria of the mother with a healthy donor.

The technique was to take a healthy donor egg, remove the (cell) nucleus and replace it with that from the mother. Scientists then fertilised the egg with the father’s sperm and implanted it into the mother’s womb. While the majority of the genetic material is from the mother and father, those from the mitochondria are from the donor, thus making her the ‘third parent’.

World’s first baby born with New “3-parent” Technique: New Scientist
The three-parent baby technique could create babies at risk of severe disease: MIT Technology Review

Three-parent baby Read More »

Post Season Begins

So, the NBA postseason 2023 starts in a couple of days. Let’s look at how the teams performed in the regular season.

n_data <- read.csv("./nba23.csv")

Most and least win

win_data <- n_data[order(-n_data$W),c(1,3)]
as_tibble(win_data)

Here are the top 10 and bottom ten

Most and least points per game

win_data_top <- n_data[order(-n_data$PTS),c(1,7)]
as_tibble(win_data_top)

win_data_bot <- n_data[order(n_data$PTS),c(1,7)]
as_tibble(win_data_bot)

Post Season Begins Read More »

Accuracy and Asymmetry

Let’s develop a simple prediction technique to identify the sex of a person based on height. Here is data from 1050 participants and has the following form.

The first step is to plot them and check their distributions.

A naive way to set up the prediction is to assign everyone with height > 64 inches as male.

y_hat <- ifelse(heights$height > 64, "Male", "Female") 
mean(heights$sex == y_hat)

The answer is an impressive 83%

But how well did it predict individually?

mean(yy[heights$sex == "Male"] == y_hat[heights$sex == "Male"])
mean(yy[heights$sex == "Female"] == y_hat[heights$sex == "Female"])

For males, the accuracy is about 94% and for females, it’s only 44%. The discrepancy prompts us to look at the respective number of samples in the set.

length(heights$sex[heights$sex == "Female"])
length(heights$sex[heights$sex == "Male"])
Females are 238, and males are 812.

Accuracy and Asymmetry Read More »

Focal Point

Remember the battle of the sexes? It had two pure strategy Nash equilibria – football-football and dance-dance. And in the absence of communication, the couple could end up with those bad outcomes. So how do we prevent such results?

A focal point is a pure strategy Nash equilibrium that all the players select because of some salient feature. An example is the heads or tails game, in which two players guess, and if both pick the same, they win prizes. Results from some experiments show an overwhelming preference for heads over tails. The reason can be that it was the first choice.

In the split money game, two parties can guess any number between 0 and 100. If the sum is lower than 100, both get their respective choices. If it is greater than 100, both get nothing. The majority selects 50.

Focal Point Read More »

Keynesian Beauty Contest

Beauty Contest was the metaphor used by John Maynard Keynes in his famous 1936 work, The General Theory of Employment, Interest, and Money, to describe prices of assets in the market. Kaynes likes investment choices to a contest in a newspaper that shows 100 pictures, and the reader needs to choose six prettiest faces. The winner is the one whose preference matches the popular choice of the overall competitors.

Levels of thinking

How will a player play this game? One approach is to assign six random photos and hope it somehow connects with the average choice. It is zero-level thinking. The first-level thinker selects the pictures she likes. The rational player knows that it is not those she likes that win prizes but the ones the others like (second-level thinking). So she wants to pick what she thinks the others would select. If you go one level up, you choose the one that others think is others’ choice.

We have seen a similar contest in an earlier post describing Richard Thaler’s experiment in the Financial Times (1997). The competition was to choose a number between 0 to 100, and the person whose choice is two-thirds of the average guess gets the prize. A first-level thinker assumes that others pick numbers at random. The average of such a collection is 50, and 2/3 of 50 is 33. The second leveller knows the other participants are rational first-levellers and would guess 33. So she will choose 2/3 x 33 = 22. Following this iterative reasoning, one will end up at the Nash Equilibrium, which is zero!

The prices

In this viewpoint, Keynes hypothesises that the market prices stocks not based on their fundamentals but on what the participants (buyers and sellers) decide.

References

Keynes, John Maynard (1936). The General Theory of Employment, Interest and Money

Keynes’s ‘beauty contest’: FT

Results from experiments

Keynesian beauty contest: Wiki

Keynesian Beauty Contest Read More »

Singing Competition

Ana, Becky and Claire are three singers entering a contest. Ana has won 4% of past competitions, Becky has 5%, and Claire has 6%. If Ana has submitted 7 albums, Becky 2 and Claire 3, what is the probability that Ana will win this time?

The general formula of Bayes’ theorem is:

\\ P(j|W) = \frac{P(W|j)*P(j)}{\sum\limits_{i = 1}^{n} P(W|i)*P(i)} \\ \\ \frac{P(W|j)*P(j)}{P(W|1)*P(1) + P(W|2)*P(2) + P(W|3)*P(3)}

In the present case, for Ana, it is:

\\ P(Ana|W) = \frac{P(W|Ana)*P(Ana)}{P(W|Ana)*P(Ana) + P(W|Becky)*P(Becky) + P(W|Claire)*P(Claire)} \\\\ \frac{0.04 *(7/12)}{0.04 *(7/12) + 0.05 *(2/12) + 0.06 *(3/12)} = 0.5

50% chance!

Singing Competition Read More »

The Dropout Fallacy

Why do people go to college? To some, it is to learn. To academics and philosophers, it is more than just learning to enrich intellectual and social capital in individuals. But to many, a college education prepares them to get a job.

And there is nothing wrong with that thought – there is a strong positive correlation between jobs and education. Here is data from the U.S.  Bureau of Labor Statistics: 

Degree Median
Salary (USD)		Unemployment
Rate (%)
Doctoral 19091.5
Professional 19241.8
Master’s15742.6
Bachlor’s13343.5
Associate’s9634.6
College, no degree8995.5
High School8096.2
Less than
High School		6268.3
Unemployment Rates and Earnings by educational attainment, 2021
Note: Data are for persons aged 25 and over. Earnings are for full-time wage and salary workers.
Source: Current Population Survey, U.S. Department of Labor, U.S. Bureau of Labor Statistics

But only until you encounter the superheroes – the Gates, the Dells and the Jobs – the college dropouts! The countless stories and speeches reinforce the theme that dropouts counterbalance their short-coming in education through their determination, superior intelligence and perseverance.

There can be a lot of factors behind the observation of successful dropouts. Foremost among them is randomness: out of the millions that have a chance but fail to complete their college, a negligible few happen to become millionaires. And they get more airtime in public. In that respect, the dropout fallacy is a survivorship bias.

The second invisible factor is related to the confounding effect of the social and cultural capital of the prosperous – such as access to a network of successful people, easy access to financing their ventures etc.

References

Unemployment Rates and Earnings by educational attainment, 2021: U.S. BLS
The Myth of the Successful College Dropout: The Atlantic

The Dropout Fallacy Read More »