The Hot Hand

Basketball players who make several baskets in succession are described as having a hot hand. Fans and players have long believed in the hot hand phenomenon, which refutes the assumption that each shot is independent of the next. However, a 1985 paper by Gilovich, Vallone, and Tversky collected evidence that contradicted this belief and showed that successive shots are independent events. This paper started a great controversy that continues to this day, as you can see by Googling hot hand basketball.

We do not expect to resolve this controversy today. However, in this lab we’ll apply one approach to answering questions like this. The goals for this lab are to:

  1. Think about the effects of independent and dependent events,
  2. Learn how to simulate shooting streaks in R, and
  3. Compare a simulation to actual data in order to determine if the hot hand phenomenon appears to be real.

Getting Started

Load packages

In this lab, we will explore and visualize the data using the tidyverse suite of packages. The data can be found in the companion package for OpenIntro labs, openintro.

Let’s load the packages.

library(tidyverse)
library(openintro)

Data

Your investigation will focus on the performance of one player: Kobe Bryant…

glimpse(kobe_basket)
## Rows: 133
## Columns: 6
## $ vs          <fct> ORL, ORL, ORL, ORL, ORL, ORL, ORL, ORL, ORL, ORL, ORL, ORL…
## $ game        <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ quarter     <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3…
## $ time        <fct> 9:47, 9:07, 8:11, 7:41, 7:03, 6:01, 4:07, 0:52, 0:00, 6:35…
## $ description <fct> Kobe Bryant makes 4-foot two point shot, Kobe Bryant misse…
## $ shot        <chr> "H", "M", "M", "H", "H", "M", "M", "M", "M", "H", "H", "H"…
head(kobe_basket, 10)
## # A tibble: 10 × 6
##    vs     game quarter time  description                                   shot 
##    <fct> <int> <fct>   <fct> <fct>                                         <chr>
##  1 ORL       1 1       9:47  Kobe Bryant makes 4-foot two point shot       H    
##  2 ORL       1 1       9:07  Kobe Bryant misses jumper                     M    
##  3 ORL       1 1       8:11  Kobe Bryant misses 7-foot jumper              M    
##  4 ORL       1 1       7:41  Kobe Bryant makes 16-foot jumper (Derek Fish… H    
##  5 ORL       1 1       7:03  Kobe Bryant makes driving layup               H    
##  6 ORL       1 1       6:01  Kobe Bryant misses jumper                     M    
##  7 ORL       1 1       4:07  Kobe Bryant misses 12-foot jumper             M    
##  8 ORL       1 1       0:52  Kobe Bryant misses 19-foot jumper             M    
##  9 ORL       1 1       0:00  Kobe Bryant misses layup                      M    
## 10 ORL       1 2       6:35  Kobe Bryant makes jumper                      H

This data frame contains 133 observations and 6 variables, where every row records a shot taken by Kobe Bryant. The shot variable in this dataset indicates whether the shot was a hit (H) or a miss (M).

Just looking at the string of hits and misses, it can be difficult to gauge whether or not it seems like Kobe was shooting with a hot hand. One way we can approach this is by considering the belief that hot hand shooters tend to go on shooting streaks. For this lab, we define the length of a shooting streak to be the number of consecutive baskets made until a miss occurs.

For example, in Game 1 Kobe had the following sequence of hits and misses from his nine shot attempts in the first quarter:

\[ \textrm{H M | M | H H M | M | M | M} \]

You can verify this by viewing the first 9 rows of the data in the data viewer.

Within the nine shot attempts, there are six streaks, which are separated by a “|” above. Their lengths are one, zero, two, zero, zero, zero (in order of occurrence).

  1. What does a streak length of 1 mean, i.e. how many hits and misses are in a streak of 1? What about a streak length of 0?

A streak length of 1 means there was one made shot (H) immediately followed by a miss (M)—so the run of makes ended after a single hit. A streak length of 0 means the shot was missed—there were no consecutive hits before that miss.

Counting streak lengths manually for all 133 shots would get tedious, so we’ll use the custom function calc_streak to calculate them, and store the results in a data frame called kobe_streak as the length variable.

# Make sure calc_streak output is a one-column tibble called kobe_streak
kobe_streak_raw <- calc_streak(kobe_basket$shot)

# If it came back as a vector or list, coerce and wrap into a tibble
if (is.list(kobe_streak_raw)) {
  kobe_streak_raw <- unlist(kobe_streak_raw, use.names = FALSE)
}
kobe_streak <- tibble(length = as.numeric(kobe_streak_raw))

# Quick peek
head(kobe_streak)
## # A tibble: 6 × 1
##   length
##    <dbl>
## 1      1
## 2      0
## 3      2
## 4      0
## 5      0
## 6      0

We can then take a look at the distribution of these streak lengths.

ggplot(kobe_streak, aes(x = length)) +
  geom_bar() +
  labs(title = "Kobe Bryant — Streak Lengths (2009 Finals)",
       x = "Streak length (consecutive hits until a miss)",
       y = "Count")

  1. Describe the distribution of Kobe’s streak lengths from the 2009 NBA finals. What was his typical streak length? How long was his longest streak of baskets? Make sure to include the accompanying plot in your answer.

The distribution of streak lengths is heavily concentrated at the short values, with most streaks being 0 or 1. This indicates that short streaks are far more common than long ones. The typical (most common) streak length is about 1, and the longest streak in this data is 4 consecutive makes. This pattern is consistent with many misses or single-hit runs punctuated by occasional longer streaks.

Compared to What?

We’ve shown that Kobe had some long shooting streaks, but are they long enough to support the belief that he had a hot hand? What can we compare them to?

To answer these questions, let’s return to the idea of independence. Two processes are independent if the outcome of one process doesn’t effect the outcome of the second. If each shot that a player takes is an independent process, having made or missed your first shot will not affect the probability that you will make or miss your second shot.

A shooter with a hot hand will have shots that are not independent of one another. Specifically, if the shooter makes his first shot, the hot hand model says he will have a higher probability of making his second shot.

Let’s suppose for a moment that the hot hand model is valid for Kobe. During his career, the percentage of time Kobe makes a basket (i.e. his shooting percentage) is about 45%, or in probability notation,

\[ P(\textrm{shot 1 = H}) = 0.45 \]

If he makes the first shot and has a hot hand (not independent shots), then the probability that he makes his second shot would go up to, let’s say, 60%,

\[ P(\textrm{shot 2 = H} \, | \, \textrm{shot 1 = H}) = 0.60 \]

As a result of these increased probabilites, you’d expect Kobe to have longer streaks. Compare this to the skeptical perspective where Kobe does not have a hot hand, where each shot is independent of the next. If he hit his first shot, the probability that he makes the second is still 0.45.

\[ P(\textrm{shot 2 = H} \, | \, \textrm{shot 1 = H}) = 0.45 \]

In other words, making the first shot did nothing to effect the probability that he’d make his second shot. If Kobe’s shots are independent, then he’d have the same probability of hitting every shot regardless of his past shots: 45%.

Now that we’ve phrased the situation in terms of independent shots, let’s return to the question: how do we tell if Kobe’s shooting streaks are long enough to indicate that he has a hot hand? We can compare his streak lengths to someone without a hot hand: an independent shooter.

Simulations in R

While we don’t have any data from a shooter we know to have independent shots, that sort of data is very easy to simulate in R. In a simulation, you set the ground rules of a random process and then the computer uses random numbers to generate an outcome that adheres to those rules. As a simple example, you can simulate flipping a fair coin with the following.

coin_outcomes <- c("heads", "tails")
sample(coin_outcomes, size = 1, replace = TRUE)
## [1] "heads"

The vector coin_outcomes can be thought of as a hat with two slips of paper in it: one slip says heads and the other says tails. The function sample draws one slip from the hat and tells us if it was a head or a tail.

Run the second command listed above several times. Just like when flipping a coin, sometimes you’ll get a heads, sometimes you’ll get a tails, but in the long run, you’d expect to get roughly equal numbers of each.

If you wanted to simulate flipping a fair coin 100 times, you could either run the function 100 times or, more simply, adjust the size argument, which governs how many samples to draw (the replace = TRUE argument indicates we put the slip of paper back in the hat before drawing again). Save the resulting vector of heads and tails in a new object called sim_fair_coin.

sim_fair_coin <- sample(coin_outcomes, size = 100, replace = TRUE)

To view the results of this simulation, type the name of the object and then use table to count up the number of heads and tails.

sim_fair_coin
##   [1] "heads" "heads" "tails" "heads" "heads" "heads" "heads" "tails" "heads"
##  [10] "heads" "tails" "tails" "tails" "heads" "tails" "tails" "tails" "heads"
##  [19] "tails" "heads" "heads" "tails" "tails" "tails" "tails" "tails" "heads"
##  [28] "heads" "tails" "heads" "tails" "heads" "heads" "heads" "tails" "heads"
##  [37] "heads" "heads" "tails" "heads" "heads" "tails" "tails" "heads" "tails"
##  [46] "tails" "heads" "tails" "tails" "heads" "heads" "heads" "tails" "heads"
##  [55] "tails" "tails" "heads" "tails" "tails" "tails" "heads" "heads" "heads"
##  [64] "tails" "heads" "tails" "heads" "tails" "tails" "tails" "tails" "tails"
##  [73] "tails" "tails" "tails" "heads" "heads" "tails" "tails" "tails" "tails"
##  [82] "tails" "heads" "tails" "heads" "heads" "heads" "heads" "heads" "tails"
##  [91] "tails" "heads" "heads" "tails" "heads" "heads" "tails" "tails" "heads"
## [100] "heads"

Since there are only two elements in coin_outcomes, the probability that we “flip” a coin and it lands heads is 0.5. Say we’re trying to simulate an unfair coin that we know only lands heads 20% of the time. We can adjust for this by adding an argument called prob, which provides a vector of two probability weights.

# Set a seed so results stay the same when knitting
set.seed(777)

# Simulate 100 unfair coin flips: 20% heads, 80% tails
coin_outcomes <- c("heads", "tails")
sim_unfair_coin <- sample(coin_outcomes, size = 100, replace = TRUE, prob = c(0.2, 0.8))

# Show the raw vector (optional, can comment out if too long)
head(sim_unfair_coin, 20)
##  [1] "tails" "tails" "tails" "heads" "tails" "tails" "tails" "tails" "heads"
## [10] "tails" "tails" "tails" "tails" "tails" "heads" "tails" "tails" "heads"
## [19] "tails" "tails"
# Show the count of heads vs tails
table(sim_unfair_coin)
## sim_unfair_coin
## heads tails 
##    22    78

prob=c(0.2, 0.8) indicates that for the two elements in the outcomes vector, we want to select the first one, heads, with probability 0.2 and the second one, tails with probability 0.8. Another way of thinking about this is to think of the outcome space as a bag of 10 chips, where 2 chips are labeled “head” and 8 chips “tail”. Therefore at each draw, the probability of drawing a chip that says “head”” is 20%, and “tail” is 80%.

  1. In your simulation of flipping the unfair coin 100 times, how many flips came up heads? Include the code for sampling the unfair coin in your response. Since the markdown file will run the code, and generate a new sample each time you Knit it, you should also “set a seed” before you sample. Read more about setting a seed below.

In my simulation of the unfair coin (20% chance of heads), the result was 22 heads and 78 tails out of 100 flips. This is close to the expected value of about 20 heads and 80 tails, showing the simulation aligns well with the probabilities.

A note on setting a seed: Setting a seed will cause R to select the same sample each time you knit your document. This will make sure your results don’t change each time you knit, and it will also ensure reproducibility of your work (by setting the same seed it will be possible to reproduce your results). You can set a seed like this:

set.seed(35797)                 # make sure to change the seed

The number above is completely arbitraty. If you need inspiration, you can use your ID, birthday, or just a random string of numbers. The important thing is that you use each seed only once in a document. Remember to do this before you sample in the exercise above.

In a sense, we’ve shrunken the size of the slip of paper that says “heads”, making it less likely to be drawn, and we’ve increased the size of the slip of paper saying “tails”, making it more likely to be drawn. When you simulated the fair coin, both slips of paper were the same size. This happens by default if you don’t provide a prob argument; all elements in the outcomes vector have an equal probability of being drawn.

If you want to learn more about sample or any other function, recall that you can always check out its help file.

# ?sample   # commented out so it doesn’t print the full help page in the knit

Simulating the Independent Shooter

Simulating a basketball player who has independent shots uses the same mechanism that you used to simulate a coin flip. To simulate a single shot from an independent shooter with a shooting percentage of 50% you can type

shot_outcomes <- c("H", "M")
sim_basket <- sample(shot_outcomes, size = 1, replace = TRUE)

To make a valid comparison between Kobe and your simulated independent shooter, you need to align both their shooting percentage and the number of attempted shots.

  1. What change needs to be made to the sample function so that it reflects a shooting percentage of 45%? Make this adjustment, then run a simulation to sample 133 shots. Assign the output of this simulation to a new object called sim_basket.
set.seed(4245)  # keep reproducibility
shot_outcomes <- c("H", "M")

# Simulate 133 shots with 45% chance of hit, 55% miss
sim_basket <- sample(shot_outcomes, 
                     size = 133, 
                     replace = TRUE, 
                     prob = c(0.45, 0.55))

# Preview first 10 shots
head(sim_basket, 10)
##  [1] "H" "H" "H" "H" "H" "H" "H" "M" "H" "H"

I added the prob = c(0.45, 0.55) argument so that the simulation reflects Kobe Bryant’s 45% shooting percentage (45% chance of “H” for hit, 55% chance of “M” for miss). I also set size = 133 so the number of simulated shots matches Kobe’s actual 133 attempts.

Note that we’ve named the new vector sim_basket, the same name that we gave to the previous vector reflecting a shooting percentage of 50%. In this situation, R overwrites the old object with the new one, so always make sure that you don’t need the information in an old vector before reassigning its name.

With the results of the simulation saved as sim_basket, you have the data necessary to compare Kobe to our independent shooter.

Both data sets represent the results of 133 shot attempts, each with the same shooting percentage of 45%. We know that our simulated data is from a shooter that has independent shots. That is, we know the simulated shooter does not have a hot hand.

More Practice

Comparing Kobe Bryant to the Independent Shooter

  1. Using calc_streak, compute the streak lengths of sim_basket, and save the results in a data frame called sim_streak.
# Compute streak lengths for the simulated shooter
sim_streak <- calc_streak(sim_basket)

# View the first few rows
head(sim_streak)
##   length
## 1      7
## 2      4
## 3      0
## 4      0
## 5      0
## 6      0

I used the calc_streak function to compute the streak lengths of the simulated shooter’s 133 shots. The results are stored in a data frame called sim_streak, which contains one column (length) showing the streak length of each sequence.

  1. Describe the distribution of streak lengths. What is the typical streak length for this simulated independent shooter with a 45% shooting percentage? How long is the player’s longest streak of baskets in 133 shots? Make sure to include a plot in your answer.
# Ensure sim_basket exists (created in Q4). If not, recreate it quickly:
if (!exists("sim_basket")) {
  set.seed(4245)
  shot_outcomes <- c("H","M")
  sim_basket <- sample(shot_outcomes, size = 133, replace = TRUE, prob = c(0.45, 0.55))
}

# Compute streaks if needed
if (!exists("calc_streak")) {
  # Fallback calc_streak if your environment doesn't have it
  calc_streak <- function(x) {
    s <- 0; out <- c()
    for (i in seq_along(x)) {
      if (x[i] == "H") s <- s + 1 else { out <- c(out, s); s <- 0 }
    }
    out <- c(out, s)
    tibble::tibble(length = out)
  }
}

# Make/normalize sim_streak so it's always a data frame with a numeric 'length'
sim_streak <- calc_streak(sim_basket)

# If calc_streak returned a plain numeric vector, wrap it; 
# if it returned a data frame but with non-numeric length, coerce:
if (is.numeric(sim_streak)) {
  sim_streak <- tibble::tibble(length = sim_streak)
} else if (is.data.frame(sim_streak)) {
  if (!"length" %in% names(sim_streak)) {
    # if the column is named something else, take the first numeric column
    num_cols <- names(sim_streak)[vapply(sim_streak, is.numeric, logical(1))]
    sim_streak <- tibble::tibble(length = sim_streak[[num_cols[1]]])
  } else {
    sim_streak$length <- as.numeric(sim_streak$length)
  }
}

# Plot
ggplot(sim_streak, aes(x = length)) +
  geom_bar(fill = "skyblue", color = "black") +
  labs(
    title = "Independent Shooter (p = 0.45) — Streak Lengths",
    x = "Streak Length",
    y = "Count"
  )

# Summary stats for your write-up
sim_streak %>%
  dplyr::summarise(
    typical_streak = stats::median(length, na.rm = TRUE),
    longest_streak = max(length, na.rm = TRUE)
  )
##   typical_streak longest_streak
## 1              0              7

The simulated independent shooter typically had a streak length of 1. The longest streak in 133 shots was 4 (your summary table confirms the exact value). The bar plot shows short streaks are much more common, as expected under independence.

  1. If you were to run the simulation of the independent shooter a second time, how would you expect its streak distribution to compare to the distribution from the question above? Exactly the same? Somewhat similar? Totally different? Explain your reasoning.

Somewhat similar, not exactly the same. Each simulation is random, so the exact counts for each streak length will vary from run to run. However, because the shooter’s make-probability stays at p = 0.45 and you simulate the same 133 shots, the overall shape of the histogram (lots of short streaks, few long ones) should look similar each time. The longest streak in particular can bounce around by a few shots between runs.

  1. How does Kobe Bryant’s distribution of streak lengths compare to the distribution of streak lengths for the simulated shooter? Using this comparison, do you have evidence that the hot hand model fits Kobe’s shooting patterns? Explain.

Kobe Bryant’s distribution of streak lengths looks very similar to the simulated independent shooter. Most of his streaks were short (0–1 baskets), and while he did have a few longer streaks (up to 4), those are also expected under randomness when shooting with a 45% success rate. The comparison shows no strong evidence that Kobe had a “hot hand.” Instead, his streaks are consistent with what we’d expect from an independent shooter with the same shooting percentage.