Lab 3 - Probability

library(tidyverse)
library(openintro)
data("kobe_basket")
data("kobe_streak")

## Warning in data("kobe_streak"): data set 'kobe_streak' not found

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 length 1 means that there was an H (Hit) followed by a M (Miss). A streak of length 0 means there were no Hs; instead, there was a M (Miss) followed by another M.

Counting streak lengths manually for all 133 shots would get tedious, so we’ll use the custom function calc_streak to calculate them, and store the results in a data frame called kobe_streak as the length variable. We can then take a look at the distribution of these streak lengths. (ggplot)

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: As seen in the data below summarizing Kobe’s streak, the distribution of Kobe’s strength lengths are more to the right, and the max streak is 4. His median, “typical” streak length was zero. See plots above and below along with summary data below.

kobe_streak <- data.frame(length = calc_streak(kobe_basket$shot))

qplot(x = length, data = kobe_streak, geom = "histogram", binwidth = 1)

summary(kobe_streak$length)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.0000  0.0000  0.0000  0.7632  1.0000  4.0000

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: Heads outcome was 15 times; a seed was set using an arbitrary number of ‘357.’ I made a mistake on the seed; The first time the code executed, I believe the heads sample was 18, not 15.

coin_results <- c("heads", "tails")
set.seed(357)
sim_unfair_coin <- sample(coin_results, size = 100, replace = TRUE, 
      prob = c(0.2, 0.8))
table(sim_unfair_coin)

## sim_unfair_coin
## heads tails 
##    15    85

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_results <- c("H", "M")
sim_basket <- sample(shot_results, size = 133, replace = TRUE,
      prob = c(0.45, 0.55))

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 <- data.frame(length = 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? Make sure to include a plot in your answer.

**Answer: The distribution of streak lengths for the simulated, independent shooter is to the right. This player’s typical, median streak length is 0 with a maximum length of 6.

qplot(x = length, data = sim_streak, geom = "histogram", binwidth = 1)

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

summary(sim_streak$length)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.000   0.000   0.000   0.942   1.000   6.000

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: If I were to run the simulation of the independent shooter a second time, I would expect the streak distribution to be similar, however I would expect it to change slightly. I would expect it to change, similar to how the change occurred above with the coin exercise. I saw variation until I set the “seed.” If running this again, I would expect the max to remain around 6 and the distribution to still be to the right, but with some variation.

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: Both streak length distributions are similar. Neither had particularly long streak lengths, indicating neither were necessarily “hot-hands” given the data sets.