You are analyzing repeated flips of a (possibly unfair) coin. ## Setup Let the probability of Heads be \(p\) (unknown in general). Assume flips are independent and identically distributed. ## Part A — Expected value for an unfair coin Define a random variable \(X\) for a single flip: - \(X = 1\) if the flip is Heads - \(X = 0\) if the flip is Tails 1. Compute \(\mathbb{E}[X]\). 2. (Optional but common follow-up) Compute \(\mathrm{Var}(X)\). ## Part B — “Getting a 3” using a geometric distribution Now flip the coin repeatedly until the first Head appears. Let \(T\) be the number of flips needed to get the first Head (so \(T\in\{1,2,3,\dots\}\)). 1. Write the distribution of \(T\) and identify it. 2. Compute \(\mathbb{P}(T=3)\) in terms of \(p\). 3. For a fair coin (\(p=0.5\)), compute the numerical value of \(\mathbb{P}(T=3)\). 4. Compute \(\mathbb{E}[T]\). ## Part C — Is the coin fair? (p-value reasoning) Suppose you ran this “flip-until-first-Head” experiment once and observed \(T=3\). You want to test: - \(H_0: p=0.5\) (fair coin) - \(H_1: p<0.5\) (coin is biased toward Tails; Heads are rarer) 1. Propose a reasonable p-value for this one observation using an appropriate tail probability under \(H_0\). 2. Briefly explain what is and is not learnable from a single observation, and what you would do instead to make the test meaningful (e.g., repeat the experiment \(n\) times).