Open Intro Probability Lab

Author

Blossom Anyanwu

Load Packages

library(tidyverse)

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.1.2     ✔ readr     2.1.4
✔ forcats   1.0.0     ✔ stringr   1.5.0
✔ ggplot2   3.4.4     ✔ tibble    3.2.1
✔ lubridate 1.9.2     ✔ tidyr     1.3.0
✔ 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

Data

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?

In a streak of 1 there is one hit and 1 miss. A streak lenght of zero indicates 1 miss and no hits at all.

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 distribution is right skewed and typically his streak length was 0. His longest streak of baskets was 4.

Simulations In R

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

[1] "tails"

# Hat analogy

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

sim_fair_coin

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

table(sim_fair_coin)

sim_fair_coin
heads tails 
   58    42

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

table(sim_unfair_coin)

sim_unfair_coin
heads tails 
   18    82

Exercise 3: In your simulation of flipping the unfair coin 100 times, how many flips came up heads?

In the simulation heads came up 24 times and tails came up 76 times.

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.

An argument would have to added to include probabilty of H as 0.45 and the probability of M as 0.55.

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

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

table(sim_basket)

sim_basket
 H  M 
48 85

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)
head(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.

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

The distribution is right skewed and the typical streak length is 0 for a simulated 45% shooting percentage. And the longest streak within this simulation 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.

I would expect it to be similar. There are 133 observations and the shooting percentages would not be changing. So the distribution would remain very similar.

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.

The simulated shooter distribution is really similar to Kobe Bryant’s. They are both right skewed and have 0 as the most common streak. As for the longest streak the simulated shooter had a length of 5 and Kobe’s was 4. Since the simulation was for an independent shooter, someone whose next shot does not depend on the previous one, I do not think the hot hand model fits Kobe.