Last lesson’s answer sheet assigned plausibility to every possible value of some quantity — but where do those numbers actually come from once real data shows up? That’s the likelihood: for a fixed candidate value of the unknown rate, how probable was the data you actually saw?
For coin-flip-shaped data (n independent flips, each landing heads with the same unknown rate p), the likelihood of seeing exactly k heads out of n flips, given a candidate rate p, is the binomial likelihood:
P(k heads in n flips | p) = C(n, k) · p^k · (1−p)^(n−k)
where C(n, k) = n! / (k!(n−k)!) counts how many orderings of k heads among n flips exist (HHHHT, HHHTH, … — the specific sequence doesn’t matter, only the count of heads, so every ordering that produces k heads adds to the same total probability).
You suspect a coin might be biased toward heads. You flip it 5 times and see 4 heads. As a first check, evaluate the likelihood of that exact data under the candidate p = 0.7 — i.e., “if the coin’s true heads-rate really were 70%, how probable was seeing 4-out-of-5?”
Compute P(4 heads in 5 flips | p = 0.7).