Probability Lab

Author

Brian Shaffer

Probability Lab

Load packages

library(tidyverse)

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.1.4     ✔ readr     2.1.5
✔ forcats   1.0.0     ✔ stringr   1.5.1
✔ ggplot2   3.4.4     ✔ tibble    3.2.1
✔ lubridate 1.9.3     ✔ tidyr     1.3.1
✔ purrr     1.0.2     
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

library(openintro)

Loading required package: airports
Loading required package: cherryblossom
Loading required package: usdata

View the data

data("kobe_basket")
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?

# A streak length of 1 consists of a hit and a miss. A streak length of 0 consists of a single 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.

# The typical streak length for Kobe or the mode is 0. The longest streak of baskets was 4. Both of these are seen in the plot below.
ggplot(data = kobe_streak, aes(x = length)) +
  geom_bar()

Compared to What?

Expected probabilities with hot hand model

P(shot 1 = H)=0.45

P(shot 2 = H|shot 1 = H)=0.60

Expected probabilities without hot hand model

P(shot 2 = H|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%.

Simulations in R

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

[1] "tails"

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

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

table(sim_fair_coin)

sim_fair_coin
heads tails 
   49    51

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

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

table(sim_unfair_coin)

sim_unfair_coin
heads tails 
   21    79

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(2378562)
sim_unfair_coin <- sample(coin_outcomes, size = 100, replace = TRUE, 
                          prob = c(0.2, 0.8))
sim_unfair_coin

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

table(sim_unfair_coin)

sim_unfair_coin
heads tails 
   23    77

# 23 flips came up heads.

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.

shot_outcomes <- c("H", "M")
sim_basket <- sample(shot_outcomes, size = 133, replace = TRUE,
                     prob = c(0.45, 0.55))
sim_basket

  [1] "M" "M" "H" "H" "H" "H" "M" "M" "H" "H" "M" "H" "H" "H" "M" "M" "M" "M"
 [19] "H" "M" "M" "M" "H" "M" "H" "M" "M" "H" "M" "M" "H" "H" "M" "H" "M" "M"
 [37] "H" "M" "H" "M" "H" "H" "M" "M" "M" "H" "M" "M" "H" "M" "M" "H" "H" "M"
 [55] "H" "M" "M" "M" "M" "H" "H" "M" "M" "H" "H" "H" "H" "H" "M" "H" "M" "M"
 [73] "H" "H" "M" "M" "H" "M" "H" "H" "M" "H" "H" "M" "H" "H" "H" "H" "M" "M"
 [91] "M" "M" "M" "M" "M" "H" "H" "H" "M" "H" "H" "H" "M" "M" "H" "H" "M" "M"
[109] "H" "M" "M" "M" "M" "M" "M" "M" "H" "M" "M" "M" "M" "M" "M" "H" "M" "M"
[127] "M" "M" "H" "M" "M" "M" "M"

table(sim_basket)

sim_basket
 H  M 
56 77

More Practice

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)

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.

# The typical streak length or mode of the independent shooter is 0.The longest streak of baskets in 133 shots was 5.
ggplot(data = sim_streak, aes(x = length)) +
  geom_bar()

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.

# It would not be exactly the same because the simulation would be run again and the sample size is small so there would be variation. The distribution should be generally the same due to the principle of an independent shooter and the probability staying the same.

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.

# Kobe Bryant's distribution matches fairly well compared to the simulated shooter. I do not think the hot hand model fits Kobe's shooting patterns or otherwise the amount of streak lengths 2 would be well higher than the simulated one not lower.