Business Context

A Meta DS is sanity-checking a toy Bayesian reasoning question used in onboarding for experimentation analysts working on Instagram Reels. Think of the coin as a hidden traffic source: one source behaves like a fair process, and one source always produces a positive signal. After observing one positive outcome, you need to infer which source was more likely selected.

Problem Statement

You randomly pick 1 of 2 coins with equal probability:

• Coin F is a fair coin
• Coin H is a double-headed coin

You flip the selected coin once and observe Heads. Determine the probability that you had picked the fair coin.

This is the same conditional-probability logic Meta DSs use when reasoning about posterior probabilities after seeing an observed event, such as an unexpectedly high early IG Save rate in a small Reels slice before checking for SRM, novelty effect, or whether CUPED-adjusted priors change interpretation.

Given Data

Requirements

1. Define the prior probabilities and likelihoods.
2. Use Bayes' theorem to compute the posterior probability of having picked the fair coin given that Heads was observed.
3. Show the total probability of observing Heads.
4. Briefly interpret why the answer is not 0.50 after seeing Heads.

Assumptions

• The coin is chosen uniformly at random.
• The observed flip is accurate.
• Only one flip is observed.
• No AARRR funnel segmentation, CUPED adjustment, or SRM issue affects the setup.