OpenIntro Probability

library(tidyverse)
library(openintro)
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"…

set.seed(271828)

Exercise 1

A streak length of 1 is a single basket followed by a miss (HM). A streak length of 0 is a consecutive miss (M).

kobe_streak <- calc_streak(kobe_basket$shot)
ggplot(data = kobe_streak, aes(x = length)) + geom_bar()

Exercise 2

Describe the distribution:

summary(kobe_streak)

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

Shape - skew right Center - median at 0, mean at 0.763 Spread - range is 4

His longest streak was a length of 4

Exercise 3

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

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))
table(sim_unfair_coin)

## sim_unfair_coin
## heads tails 
##    24    76

24 flips in the simulation came up heads

Exercise 4

Change the given function by adding a probability statement:

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 
## 56 77

Exercise 5

sim_streak <- calc_streak(sim_basket)
ggplot(data = sim_streak, aes(x = length)) + geom_bar()

### Exercise 6

summary(sim_streak)

##      length      
##  Min.   :0.0000  
##  1st Qu.:0.0000  
##  Median :0.0000  
##  Mean   :0.7179  
##  3rd Qu.:1.0000  
##  Max.   :7.0000

Describe the distribution:

Shape - skew right Center - median at 0, mean at 0.577 Spread - range of 8

The player’s longest streak is 8

Exercise 7

If I were to run the simulation again, I would expect it to be somewhat similar to this one, with the same shape and median, but a slightly different mean. I do not think the spread would remain the same, as the value 8 is an outlier. The sample size 133 is large enough to produce similar results.

Exercise 8

Kobe and the simulation’s distribution have similar shapes and centers. Although Kobe’s distribution features relatively higher frequencies at 1 and 3 for streak length, I do not believe that this is compelling evidence that the hot-hand hypothesis has any significance. I would have to do analysis such as the chi-square test to determine the significance in any quantitative sense. Another issue is that our simulation could simply be assuming the wrong probability of a hit given Kobe’s distribution. A binomial probability fit test would help to determine if this is an issue.