Lab 3: Probability

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) to compare a simulation to actual data in order to determine if the hot hand phenomenon appears to be real.

Getting started

library(tidyverse)
library(openintro)

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"…

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?

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.

summary(kobe_streak)

##      length      
##  Min.   :0.0000  
##  1st Qu.:0.0000  
##  Median :0.0000  
##  Mean   :0.7632  
##  3rd Qu.:1.0000  
##  Max.   :4.0000

barplot(table(kobe_streak))

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.

set.seed(112490)

results <- c("heads", "tails")
simulation_unfair_coin <- sample(results, size = 100, replace = TRUE, prob = c(0.3, 0.7))
table(simulation_unfair_coin)

## simulation_unfair_coin
## heads tails 
##    28    72

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.

set.seed(091320)
possible_outcomes <- c("H", "M")

sim_basket <- sample(possible_outcomes, size = 133, replace = TRUE, 
                       prob = c(0.45, 0.55))
table(sim_basket)

## sim_basket
##  H  M 
## 53 80

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 <- calc_streak(sim_basket)
table(sim_streak)

## length
##  0  1  2  3  4 
## 51 16  7  5  2

barplot(table(sim_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.

set.seed(09152020)
possible_outcomes_1<-c("H", "M")
sim_basket<-sample(possible_outcomes_1, size=133, replace=TRUE,  prob=c(0.45, 0.55))
sim_streak<-calc_streak(sim_basket)
barplot(table(sim_streak))

We’ve once again set the seed to 09152020 and generated possible outcomes of H for hit and M for miss. The number of shots is set to 133, and we’ve adjusted the probability to reflect a shooter with a 45% success rate. The player’s average streak length is 0, while their longest streak over 133 shots is 3. From the graph, we can observe that the distribution is unimodal and skewed to the right.

table(sim_streak)

## length
##  0  1  2  3 
## 37 20 11  6

max(sim_streak)

## [1] 3

min(sim_streak)

## [1] 0

summary(sim_streak)

##      length      
##  Min.   :0.0000  
##  1st Qu.:0.0000  
##  Median :0.5000  
##  Mean   :0.8108  
##  3rd Qu.:1.0000  
##  Max.   :3.0000

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.

I believe that if we run the independent shooter simulation again, since the probability is set to reflect a 45% shooting success rate, the results should be similar. This is a key point, as the most common streak would still likely be 0.

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.

barplot(table(kobe_streak))

barplot(table(sim_streak))

The streak length distributions for both the simulation and Kobe Bryant’s shooting data are largely comparable. I don’t believe there is enough evidence to support the “hot hand” theory in Kobe’s case. Both data sets show right-skewed plots. However, one key difference is that Kobe had a longest streak of 4, while the simulated player had a longest streak of 3, which could be due to chance. In both cases, the most common streak length was 0, suggesting, as previously mentioned, that Kobe’s shots seem to be independent of each other.