library(tidyverse)
library(openintro)Probability Lab
Probability
The Hot Hand
Getting Started
Load Packages
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"…data("kobe_basket")Excercise 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 of 1 means one shot was made. A streak of one has one hit. A streak of 0 means no shots were made.
kobe_streak <- calc_streak(kobe_basket$shot)
ggplot(data = kobe_streak, aes(x = length)) +
geom_bar()
Excercise 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: Most of Kobe’s streaks were 0 streaks with his longest streak being 4 makes in a row. The longest streak of baskets only happened once as shown in the plot above.
Compared to What?
Two processes are independent if the outcome of one doesn’t effect the outcome of the other
P (shot 1 = H) = 0.45 If not independent: P(shot 2 = H | shot 1 = H) = 0.60 If independent: P(shot 2 = H | shot 1 = H) = 0.45
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] "heads" "tails" "tails" "tails" "tails" "tails" "heads" "tails" "heads"
[10] "heads" "tails" "tails" "heads" "tails" "tails" "tails" "tails" "heads"
[19] "heads" "tails" "heads" "tails" "tails" "heads" "tails" "tails" "tails"
[28] "tails" "tails" "tails" "heads" "tails" "heads" "heads" "heads" "tails"
[37] "heads" "tails" "tails" "tails" "tails" "tails" "tails" "tails" "heads"
[46] "tails" "heads" "heads" "heads" "tails" "tails" "heads" "heads" "tails"
[55] "tails" "heads" "tails" "heads" "heads" "heads" "heads" "heads" "tails"
[64] "tails" "tails" "heads" "tails" "heads" "tails" "heads" "heads" "heads"
[73] "heads" "heads" "tails" "tails" "tails" "tails" "heads" "tails" "tails"
[82] "heads" "heads" "tails" "heads" "tails" "heads" "heads" "tails" "heads"
[91] "heads" "tails" "tails" "tails" "tails" "heads" "tails" "tails" "heads"
[100] "heads"table(sim_fair_coin)sim_fair_coin
heads tails
44 56 sim_unfair_coin <- sample(coin_outcomes, size = 100, replace = TRUE,
prob = c(0.2, 0.8))
sim_unfair_coin [1] "tails" "tails" "tails" "heads" "tails" "heads" "tails" "tails" "tails"
[10] "heads" "tails" "tails" "heads" "tails" "tails" "tails" "tails" "tails"
[19] "tails" "tails" "tails" "tails" "tails" "tails" "tails" "tails" "heads"
[28] "tails" "tails" "tails" "tails" "tails" "tails" "tails" "tails" "tails"
[37] "heads" "tails" "tails" "tails" "tails" "tails" "tails" "tails" "tails"
[46] "heads" "tails" "heads" "tails" "tails" "heads" "heads" "tails" "heads"
[55] "tails" "tails" "tails" "heads" "tails" "tails" "heads" "tails" "tails"
[64] "tails" "heads" "heads" "tails" "tails" "tails" "tails" "heads" "heads"
[73] "tails" "tails" "heads" "tails" "tails" "tails" "tails" "tails" "tails"
[82] "tails" "tails" "tails" "heads" "heads" "tails" "tails" "tails" "tails"
[91] "tails" "tails" "tails" "tails" "tails" "tails" "tails" "heads" "heads"
[100] "tails"table(sim_unfair_coin)sim_unfair_coin
heads tails
22 78 Excercise 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.
Answer: Heads came up 15 times. sim_unfair_coin <- sample(coin_outcomes, size = 100, replace = TRUE, prob = c(0.2, 0.8))
Simulating the Independent Shooter
shot_outcomes <- c("H", "M")
sim_basket <- sample(shot_outcomes, size = 1, replace = TRUE)
sim_basket[1] "M"table(sim_basket)sim_basket
M
1 Excercise 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: Have to add a probability argument and change the size to 133.
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
63 70 More Practice
Excercise 5: Using calc_streak, compute the streak lengths of sim_basket, and save 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? Include plot.
ggplot(sim_streak, aes(x=length)) +
geom_bar()
Answer: The plot is skewed. The typical streak length is 0 and the longest streak of baskets is 5.
Exercise 7: If you were to run the simulation again, how would you expect its streak distribution to compare to the distribution above? Exactly the same? Somewhat similar? Totally different? Explain
Answer: I would expect the streak distribution to be very similar with the same part being skewed even the count in each streak would probably change.
Exercise 8: How does Kobe Bryants’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: Kobe and the simulated shooter had similarities in the fact that they were skewed but the simulated shooter had more of a gradual decrease while Kobe had pretty similar counts in the length of 2 and 3. Using this comparison we can see that the hot hand model somewhat fits Kobe’s shooting pattern as they match up pretty similarly. Except for the fact that Kobe did not have a 5 length streak at any point.