---
title: "Session 10 — Bernoulli intuition"
author: "Prof. Dae-Jin Lee (daelee@faculty.ie.edu)"
date: "Fall 2026"
output:
  html_document:
    theme: flatly
---

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

One trial success probability $p=0.35$ (e.g., a randomly selected proposal passes a committee).

```{r}
set.seed(10)
x <- rbinom(5000, size = 1, prob = 0.35)
c(mean = mean(x), var = var(x))
c(theory_mean = 0.35, theory_var = 0.35 * 0.65)
```

**Interpret:** simulated mean ≈ $p$; simulated variance ≈ $p(1-p)$.