Probability Lab

Author

Lydia Baick

Probability

The Hot Hand

Getting Started

Load Packages

library(tidyverse)
library(openintro)

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")

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 of one means one single hit. A streak of zero means a miss

kobe_streak <- calc_streak(kobe_basket$shot)

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

Excerise 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 = His typical streak length was 0. His longest streak length of baskets was 4.

Compared to What

Simulations in R

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

[1] "heads"

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

sim_fair_coin

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

table(sim_fair_coin)

sim_fair_coin
heads tails 
   42    58

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

set.seed(10192007)

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

table(sim_unfair_coin)

sim_unfair_coin
heads tails 
   20    80

Exercise 2

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 = After simulating the flipping of 100 unfair coins 13 of the flips were heads. The code I used for sampling the unfair coing was: sim_unfair_coin table(sim_unfair_coin)

Simulaing the Independent Shooter

shot_outcomes <- c("H", "M")
sim_basket <- sample(shot_outcomes, size = 1, replace = TRUE)

sim_basket <- sample(shot_outcomes, size = 100, replace = TRUE)

sim_basket

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

table(sim_basket)

sim_basket
 H  M 
54 46

sim_basket <- sample(shot_outcomes, size = 133, replace = TRUE, prob = c(.45, .55))

sim_basket

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

table(sim_basket)

sim_basket
 H  M 
56 77

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 = To sample the function so that it reflects a shooting percentage of 45% we need to add an argument called prob.

Excerise 5

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

Answer = code below

sim_streak <- calc_streak(sim_basket)

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

Excerise 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 for a shooter with a 45% shooting percentage is a streak of zero. The longest streak of baskets in 133 shots is 7.

Excerise 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 = The streak distribution would be somewhat simlilar because the percentage of hits would always be the same, 45%. So I would not be surprised if the streak distribution was that same or very similar to the first run.

Excerise 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 = I think that Kobe’s distribution of streak lengths is similar to the distribution of streak lengths for the simulated shooter because for both, the typical streak length was a streak of 0 and the count for 2, 3, and 4 seem pretty low. I don’t think the hot hand model fits Kobe’s shooting patterns because to me, both distribution of streak lengths seem pretty similar proving that there is no hot hand.