P(4 heads in 5 flips | p = 0.7) = C(5,4) · 0.7⁴ · 0.3¹ = 5 × 0.2401 × 0.3 ≈ 0.360.
Walking the formula: C(5,4) = 5 (there are 5 different flip-sequences with exactly one tail — TAILS-then-4-heads, or the tail in any of the other 4 positions). 0.7⁴ = 0.2401 is the probability of any one specific 4-heads outcome; 0.3¹ is the probability that the remaining flip is a tail. Multiply all three together and about 36% of the time, a truly-70%-heads coin produces exactly this data over 5 flips.
The number alone answers nothing yet — it only becomes useful in comparison. 0.360 isn’t “the probability the coin is biased” and it isn’t a verdict on p = 0.7 in isolation; it’s one entry on an answer sheet like last lesson’s, one likelihood value attached to one candidate rate. The move that makes it Bayesian: compute the same likelihood formula under other candidate rates too — say p = 0.5 (a fair coin) gives C(5,4)·0.5⁴·0.5¹ = 5 × 0.03125 ≈ 0.156, noticeably lower. Line up enough candidate p values side by side, each with its own likelihood, and you’ve built the entire likelihood curve — the second ingredient (alongside a prior over p) that a posterior distribution multiplies together, point by point, exactly the way odds-form multiplied a single likelihood ratio into prior odds back in Stage 1.
Why the binomial shape specifically: it applies whenever data is a count of successes out of a fixed number of independent identical trials, each with the same unknown success probability — coin flips, yes, but also click-through counts, defect counts in a batch, or “how many users converted out of those shown a variant.” Any time that shape shows up, this same formula is the likelihood machinery underneath.
Where this goes: next lesson puts a prior over p (a starting answer sheet before any data) next to this likelihood and multiplies them together, point by point across every candidate rate — producing the posterior answer sheet: beta-binomial updating, the first fully worked distribution-level Bayesian update in this track.