Probability Lab

Author

Shi Ji

Probability Lab

load 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 length of 1 means 1 hit followed by 1 miss. A streak length of zero means 1 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: The typical streak length is 0, and the longest streak of baskets is 4.

Simulations in R

coin_outcomes <- c("heads", "tails")
sample(coin_outcomes, size = 1, replace = TRUE)
[1] "tails"
Simulate flipping a fair coin 100 times
sim_fair_coin <- sample(coin_outcomes, size = 100, replace = TRUE)
To view the results of this simulation
sim_fair_coin
  [1] "tails" "tails" "tails" "tails" "heads" "tails" "heads" "tails" "tails"
 [10] "tails" "heads" "tails" "heads" "tails" "tails" "tails" "tails" "tails"
 [19] "tails" "heads" "tails" "tails" "tails" "heads" "heads" "tails" "tails"
 [28] "heads" "heads" "heads" "tails" "heads" "tails" "heads" "heads" "heads"
 [37] "heads" "tails" "heads" "tails" "tails" "heads" "tails" "tails" "tails"
 [46] "heads" "heads" "heads" "tails" "heads" "tails" "heads" "heads" "tails"
 [55] "heads" "heads" "heads" "tails" "tails" "tails" "heads" "tails" "tails"
 [64] "heads" "heads" "heads" "heads" "heads" "tails" "heads" "heads" "tails"
 [73] "tails" "tails" "heads" "heads" "heads" "tails" "tails" "tails" "heads"
 [82] "heads" "heads" "tails" "tails" "heads" "tails" "tails" "tails" "tails"
 [91] "tails" "heads" "heads" "tails" "heads" "heads" "tails" "tails" "heads"
[100] "heads"
table(sim_fair_coin)
sim_fair_coin
heads tails 
   47    53
Adding an argumen provides a vector of two probability weights
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.

set.seed(1842)
coin_outcomes <- c("heads", "tails")
sim_fair_coin <- sample(coin_outcomes, size = 100, replace = TRUE)
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 
   21    79 
# Answer: 21 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))
table(sim_basket)
sim_basket
 H  M 
57 76

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)
sim_streak
   length
1       1
2       0
3       0
4       2
5       0
6       0
7       0
8       0
9       1
10      3
11      0
12      2
13      6
14      3
15      0
16      1
17      0
18      0
19      1
20      0
21      1
22      0
23      0
24      0
25      0
26      1
27      0
28      0
29      0
30      1
31      2
32      2
33      0
34      0
35      1
36      1
37      0
38      0
39      0
40      0
41      3
42      0
43      0
44      2
45      0
46      1
47      0
48      1
49      0
50      0
51      1
52      2
53      0
54      2
55      0
56      0
57      0
58      0
59      1
60      1
61      0
62      0
63      1
64      1
65      2
66      0
67      0
68      1
69      0
70      0
71      2
72      0
73      2
74      1
75      3
76      1
77      0

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.

barplot(table(sim_streak), col = "lightyellow")

# Answer: 
# The distribution is right skewed. 
# The typical streak lenght is 0.
# The longest streak of baskets is lenght 6.

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 
# That would be some differences since the sample is random, 
# but the results would not be that much different.
# However, here doesn't included any seed values.
# If we set some different seed values, chances will be appear in the result.

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
# Because the distributions look very similar. 
# Therefore, here is no evidence showing Kobe Bryant’s hot hand shooting patterns.