DATA 606 Lab 3 - Probability

library(tidyverse)
library(openintro)

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? Streak of one mean Kobe Bryant hit a shot then missed the next shot In a streak of 1 there are one hit and one miss

Streak of 0 means Kobe Bryant missed one shot. There are cases where he missed one and then again missed.

kobe_streak <- calc_streak(kobe_basket$shot)

We can then take a look at the distribution of these streak lengths.

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. Kobe Bryant streak distribution is right skewed and unimodal(one peak). Typical streak length was 0 and longest streak length was 4(longest streak of baskets).

summary(kobe_streak)

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

barplot(table(kobe_streak))

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. I decided to use a probability of 0.3 and 0.7 to test out the unfair coin flips. In my example, I set seed to my birth date in order to prevent unique outcomes when knitting. Heads appeared 28 times and Tails appeared 72.

set.seed(112490)

results <- c("heads", "tails")
simulation_unfair_coin <- sample(results, size = 100, replace = TRUE, prob = c(0.3, 0.7))
table(simulation_unfair_coin)

## simulation_unfair_coin
## heads tails 
##    28    72

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. Once again for the purpose of lab 3 we’ve set a seed[to current date randomly]. Then we stated the possible outcomes as H for hit and M for miss. Using prob, we changed the probability of hits to .45 and misses to .55

set.seed(091320)
possible_outcomes <- c("H", "M")

sim_basket <- sample(possible_outcomes, size = 133, replace = TRUE, 
                       prob = c(0.45, 0.55))
table(sim_basket)

## sim_basket
##  H  M 
## 53 80

Exercise 5

Using calc_streak, compute the streak lengths of sim_basket, and save the results in a data frame called sim_streak. As outlined, we’ve computed the streak lengths of sim_basket by utilizing a data frame name sim_streak and viewing output/barplot. We see with reviewing the sim_basket data as is, it shows unimodal right skewed. Typical streak length is 0 and longest streak length is 4

sim_streak <- calc_streak(sim_basket)
table(sim_streak)

## sim_streak
##  0  1  2  3  4 
## 51 16  7  5  2

barplot(table(sim_streak))

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. Once again, we’ve set a seed to 09152020. we’ve created the possible outcomes of H for hit and M for miss. We’ve set the size to 133 indicating the number of shots. Using prob, we’ve adjusted the probability of the data to reflect shooter with 45% shooting range. The players typical streak length is 0. Players longest streak length in 133 shots is 3. By the graphic we see this is again unimodal right skewed.

set.seed(09152020)
possible_outcomes_1<-c("H", "M")
sim_basket<-sample(possible_outcomes_1, size=133, replace=TRUE,  prob=c(0.45, 0.55))
sim_streak<-calc_streak(sim_basket)
barplot(table(sim_streak))

table(sim_streak)

## sim_streak
##  0  1  2  3 
## 37 20 11  6

max(sim_streak)

## [1] 3

min(sim_streak)

## [1] 0

summary(sim_streak)

##      length      
##  Min.   :0.0000  
##  1st Qu.:0.0000  
##  Median :0.5000  
##  Mean   :0.8108  
##  3rd Qu.:1.0000  
##  Max.   :3.0000

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. My opinion is that if we run the simulation of the independent shooter a second time, being that we’ve set the prob to reflect a 45% successful shooting range, the data should be similar. This is a very important detail. The most typical streak would still be 0. The only possible variation would possibly be the longest length of streaks[which would still be low].

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.

Overall the distribution of streak lenghths in comparison for sim and kobe are similar. I would suggest that there is not enough evidence that supports the ‘hot hand’ theory to kobe bryants shooting data. Some similarities we notice are that the data plots are right skewed.One clear distinction we do see however is that Kobe bryant had a longest streak length of 4 in our comparison to the sim player which had 3 for longest length of streak. This could be by chance as well. Both sim and Kobe bryant have a most common streak length of 0. As suggested before, Kobe Bryants shots appear to be independent of each other.

barplot(table(kobe_streak))

barplot(table(sim_streak))