Derive Probability of Even Sum in Bernoulli Trials

Q: Derive Probability of Even Sum in Bernoulli Trials

The question evaluates probabilistic reasoning and discrete stochastic-process concepts, specifically parity of sums of i.i.d. Bernoulli(p) random variables and expected waiting times for consecutive outcomes in fair coin flips.

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Question

Probability Puzzles: Parity and Runs

Context

You are given two independent probability problems commonly seen in data-science take-home assessments. Assume all random variables are defined on the same probability space.

Problems

Parity of a Bernoulli Sum
- Let X1, ..., Xn be i.i.d. Bernoulli(p). Let S = Σ_{i=1}^n X_i.
- Derive a closed-form expression for P(S is even).
Waiting Time to First Repeat (Fair Coin)
- A fair coin is flipped until two consecutive identical outcomes (HH or TT) appear.
- Find the expected number of flips required.

Derive Probability of Even Sum in Bernoulli Trials

Probability Puzzles: Parity and Runs

Context

Problems

Solution

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