---
title: "Session 9 — Simulation intuition"
subtitle: "OpenIntro / QSS spirit | Probability and Statistics (MIR)"
author: "Prof. Dae-Jin Lee (daelee@faculty.ie.edu)"
date: "Fall 2026"
output:
  html_document:
    toc: true
    toc_float: true
    theme: flatly
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, message = FALSE, warning = FALSE)
library(tidyverse)
```

Suppose 30% of voters support a diplomatic initiative. Simulate Bernoulli outcomes.

```{r}
set.seed(9)
support <- rbinom(1000, size = 1, prob = 0.30)
mean(support)
```

# Stability grows with more trials

```{r}
set.seed(9)
c(
  n20 = mean(rbinom(20, 1, 0.30)),
  n200 = mean(rbinom(200, 1, 0.30)),
  n20000 = mean(rbinom(20000, 1, 0.30))
)
```

**Interpret:** each run is random; larger $n$ usually lands closer to 0.30.

# Many replications of a small sample

```{r}
set.seed(9)
many_small <- replicate(500, mean(rbinom(20, 1, 0.30)))
tibble(est = many_small) |>
  ggplot(aes(x = est)) +
  geom_histogram(bins = 20) +
  labs(
    title = "Sampling noise with n = 20",
    x = "Estimated support share",
    y = "Count"
  )
```

Small-$n$ IR “rates” can swing a lot — ask for the sample size behind any probability claim.