Probability Lab

Author

J Chris

Probability Lab

Load the libraries

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     1.3.0
✔ 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

Access 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?

Answer: A streak implies there is one hit and one miss. The miss is needed because that terminates the streak leaving behind only one hit.

Bar graph of Kobe’s streaks

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: It seems that apart from 0, his typical streak length was 1. Kobe’s longest streak of baskets was 4.

Coin Flip

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

table(sim_fair_coin)

sim_fair_coin
heads tails 
   57    43

set.seed(12112000)

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

sim_unfair_coin

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

table(sim_unfair_coin)

sim_unfair_coin
heads tails 
   13    87

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: In my simulation head came up only 13 times out of the 100 total flips. ( set.seed(12112000)

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

sim_unfair_coin table(sim_unfair_coin))

Simulating the independent shooter

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

set.seed(46290)

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

Independent_shooter_streak

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

table(Independent_shooter_streak)

Independent_shooter_streak
 H  M 
58 75

# Exercise #5
sim_streak <- calc_streak(Independent_shooter_streak)

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

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: The function or parameter prob() needs to be added to the sample function so that you can choose a percentage of 45%.

Exercise #5

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

Answer: Done.

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.

Answer: The typical streak length apart from 0 was also 1 from the simulated shooter. The longest streak for the player is 5 baskets. Another interesting observation is that there were no streaks of 4.

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: I believe the streak distribution would be very similar because we gave the shooter a target shooting percentage and the same 133 shots it can take.

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: Based on the first simulation I ran for the simulated shooter, the two bar graphs show us similar results. Both graphs are right-skewed and the typical streak for both is 1. However, the graphs do differ when it comes to the the longest streak being one basket higher for the simulated shooter (5 baskets). Kobe’s graph shows more consistency for the amount of times the streaks were made though. The graph of the simulated shooter has many more streaks of zero. This leads me to believe that the hot hand model somewhat fits Kobe’s shooting patterns in the aspect of consistency. It might have appeared like Kobe had a hot hand because he could consistently hit more streaks of one, atleast compared to this simulated shooter. Kobe’s streaks of two and three were basically the same if not identical.