---
title: "Lab 3: Probability, Independence, and Simulation"
author: "Your name"
format: html
editor: visual
editor_options:
  chunk_output_type: console
---

## Setup

```{r}
library(dplyr)
library(ggplot2)

load(url("https://raw.githubusercontent.com/GarciaRios/govt_3990/gh-pages/Labs/lab3/resources/hot_hand.RData"))
```

## Exercise 1: Estimate probability from observed frequency

```{r}
kobe_basket %>%
  count(shot) %>%
  mutate(prop = n / sum(n))

p_hit <- mean(kobe_basket$shot == "H")
p_hit
```

What is the observed probability of a hit?

> 

What is the observed probability of a miss?

> 

Check that the two probabilities sum to 1.

```{r}

```

## Exercise 2: Conditional probability

Create a variable containing the previous shot.

```{r}
kobe_transitions <- kobe_basket %>%
  group_by(game) %>%
  mutate(previous_shot = lag(shot)) %>%
  ungroup() %>%
  filter(!is.na(previous_shot))
```

Calculate the conditional proportions.

```{r}
kobe_transitions %>%
  count(previous_shot, shot) %>%
  group_by(previous_shot) %>%
  mutate(prob = n / sum(n))
```


Visualize the same comparison with a proportional bar chart.

```{r}
ggplot(kobe_transitions, aes(x = previous_shot, fill = shot)) +
  geom_bar(position = "fill") +
  labs(
    x = "Previous shot",
    y = "Proportion of next shots",
    fill = "Current shot",
    title = "What happens after a hit or miss?"
  ) +
  theme_minimal()
```

What is the observed value of $P(H_t \mid H_{t-1})$?

> 

What is the observed value of $P(H_t \mid M_{t-1})$?

> 

Are the two conditional probabilities exactly the same?

> 

If shots were independent, what relationship would we expect between them and the overall probability of a hit?

> 

What does the proportional bar chart show about the chance of a hit after a previous hit versus after a previous miss?

> 

Does a difference in this one sample automatically prove dependence? Why not?

> 

## Exercise 3: Simulate an independent shooter

Set a seed.

```{r}
set.seed(311)
```

Simulate the same number of shots, using the observed hit probability.

```{r}
shot_outcomes <- c("H", "M")

sim_basket <- sample(
  shot_outcomes,
  size = nrow(kobe_basket),
  replace = TRUE,
  prob = c(p_hit, 1 - p_hit)
)
```

Check the simulated shooting percentage.

```{r}
mean(sim_basket == "H")
table(sim_basket)
```

Create transitions and calculate conditional probabilities.

```{r}
sim_transitions <- tibble(shot = sim_basket) %>%
  mutate(previous_shot = lag(shot)) %>%
  filter(!is.na(previous_shot))

sim_transitions %>%
  count(previous_shot, shot) %>%
  group_by(previous_shot) %>%
  mutate(prob = n / sum(n))
```

Compare $P(H_t \mid H_{t-1})$ and $P(H_t \mid M_{t-1})$ in the simulation.

> 

Are they exactly equal?

> 

We know the simulated shots are independent. Why can the sample conditional probabilities still differ?

> 

## Exercise 4: Compare streaks

Calculate and plot Kobe's streak lengths.

```{r}
kobe_streak <- calc_streak(kobe_basket$shot)

ggplot(kobe_streak, aes(x = length)) +
  geom_histogram(binwidth = 1, boundary = -0.5) +
  labs(
    x = "Streak length",
    y = "Count",
    title = "Observed shooting streaks"
  ) +
  theme_minimal()
```

Summarize the observed streaks.

```{r}
kobe_streak %>%
  summarise(
    mean_streak = mean(length),
    median_streak = median(length),
    max_streak = max(length)
  )
```

Now calculate and plot the simulated streaks.

```{r}
sim_streak <- calc_streak(sim_basket)

ggplot(sim_streak, aes(x = length)) +
  geom_histogram(binwidth = 1, boundary = -0.5) +
  labs(
    x = "Streak length",
    y = "Count",
    title = "Independent-shooter simulation"
  ) +
  theme_minimal()
```

```{r}
sim_streak %>%
  summarise(
    mean_streak = mean(length),
    median_streak = median(length),
    max_streak = max(length)
  )
```

Compare the observed and simulated streak distributions.

Are long streaks possible under independence?

> 

How do the typical and maximum streak lengths compare?

> 

Why would one simulated sequence not be enough to make a strong conclusion about the hot-hand question?

> 

## On Your Own

### A second independent-shooter simulation

Choose a new seed.

```{r}

```

Simulate another independent shooter with the same number of shots and the same overall hit probability, `p_hit`.

```{r}

```

Calculate the simulated shooting percentage.

```{r}

```

Calculate $P(H_t \mid H_{t-1})$ and $P(H_t \mid M_{t-1})$.

```{r}

```

Calculate the maximum streak length.

```{r}

```

Compare your second simulation to the first simulation and to the observed Kobe data in 3–4 sentences. Explain why differences between simulations do not imply that the probability model changed.

>