Compute expectations and test fairness for coin flips

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).