Probability The Hot Hand Lab

The Hot Hand

library(tidyverse)

## -- Attaching packages --------------------------------------------------------------------------------------- tidyverse 1.3.0 --

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## v tibble  3.0.3     v dplyr   1.0.2
## v tidyr   1.1.2     v stringr 1.4.0
## v readr   1.3.1     v forcats 0.5.0

## -- Conflicts ------------------------------------------------------------------------------------------ tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()

library(openintro)

## Loading required package: airports

## Loading required package: cherryblossom

## Loading required package: usdata

Data

Your investigation will focus on the performance of one player: Kobe Bryant of the Los Angeles Lakers. His performance against the Orlando Magic in the 2009 NBA Finals earned him the title Most Valuable Player and many spectators commented on how he appeared to show a hot hand. The data file we’ll use is called kobe_basket.

glimpse(kobe_basket)

## Rows: 133
## Columns: 6
## $ vs          <fct> 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...
## $ quarter     <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3...
## $ time        <fct> 9:47, 9:07, 8:11, 7:41, 7:03, 6:01, 4:07, 0:52, 0:00, 6...
## $ description <fct> Kobe Bryant makes 4-foot two point shot, Kobe Bryant mi...
## $ shot        <chr> "H", "M", "M", "H", "H", "M", "M", "M", "M", "H", "H", ...

Exercise 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?

Answer: A streak length of 1 means that he made 1 shot and then missed the next shot.In streak 1 he had 1 shit and 1 miss. Streak 0 means that he had misses and continued to miss.

kobe_streak <- calc_streak(kobe_basket$shot)

ggplot(data = kobe_streak, aes(x = length)) +
  geom_bar()

Exercise 2

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.

Answer: According to the data on the bar. His typical streak length was 0 and his longest streak of baskets was 4.

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"

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

sim_fair_coin

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

table(sim_fair_coin)

## sim_fair_coin
## heads tails 
##    41    59

sim_unfair_coin <- sample(coin_outcomes, size = 100, replace = TRUE, 
                          prob = c(0.2, 0.8))

Exercise 3

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.

Answer: Heads came up 48 times and tails 52 times. In the unfair coin toss, heads came out to be 26 and tails was 74.

set.seed(090392)
coin_outcomes <-c("heads","tails")
sample(coin_outcomes, size = 1, replace = TRUE)

## [1] "tails"

sim_unfair_coin <- sample(coin_outcomes, size = 100, replace = TRUE, 
                          prob = c(0.2, 0.8))
table(sim_unfair_coin)

## sim_unfair_coin
## heads tails 
##    22    78

Simulating the Independent Shooter

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

Exercise 4

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.

Answer: There would be 64 hits and 69 misses.

shot_outcomes <- c("H", "M")
sim_basket <- sample(shot_outcomes, size = 133, replace = TRUE)
sim_basket <- sample(shot_outcomes, size = 133, replace = TRUE,
                     prob = c(0.45, 0.50))
table(sim_basket)

## sim_basket
##  H  M 
## 70 63

Exercise 5

Using calc_streak, compute the streak lengths of sim_basket, and save the results in a data frame called sim_streak.

sim_streak <- sim_basket %>%
  calc_streak()

Exercise 6

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.

ggplot(data = sim_streak, aes(x = length)) +
  geom_bar()

Answer: The typical streak length is 0, based on the graph and the longest streak the player had is 5.

Exercise 7

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.

Answer: If I was to run the simulation again. I would honestly think the simulation would be pretty similar. Reason why being because each shots are individual from each other. Probability in the simulation should not affect whether the player would make the shot between both simulations.

exercise 8

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.

Answer: Kobe Bryant’s and the simulation shooters streak lenth look pretty similar. Both of them have a typical streak of 0 and also both of distributions skew to the right to a similar range. Overall both are similar. Both distributions are similar and because of that there isn’t enough evidence to say the hot hand model fits Kobe Byrant’s shooting pattern.