What is the difference between probability and likelihood after seeing 7 heads in 10 coin flips?

## One formula, two readings Take the standard model for coin flips: the coin lands heads with probability `p`, flips are independent, and you record how many heads appear in `n` flips. The model says `P(k heads in n flips | p) = C(n, k) * p^k * (1-p)^(n-k)` where `C(n, k)` counts the orderings of k heads among n flips. With `n = 10` and `k = 7` this is `120 * p^7 * (1-p)^3`. The expression on its own does not tell you which symbol is the variable and which is held fixed. That choice is the entire distinction between a probability and a likelihood. ### Probability: parameter fixed, data varying Pick a value of the parameter, say `p = 0.5`, and let the data range. You now have a function of `k` over the values 0 through 10. It is a genuine probability distribution: every value is non-negative and the eleven values add to 1. This is the reading you use when you ask forward questions — what outcomes should I expect if the coin behaves like this? ### Likelihood: data fixed, parameter varying Now do the opposite. The data are in hand and no longer uncertain: you saw 7 heads. Let `p` range over the interval from 0 to 1 and write `L(p) = 120 * p^7 * (1-p)^3`. This is the likelihood function. It answers a backward question — for each candidate value of the bias, how well does that value account for what actually happened? It is highest at `p = 0.7` and falls away on either side, quickly toward 0 and 1 where the observed mix of heads and tails becomes implausible. ### Why the likelihood is not a distribution over p Two reasons, and interviewers probe both. First, arithmetic. Integrating `120 * p^7 * (1-p)^3` over p from 0 to 1 gives `1/11`, not 1. (For a binomial likelihood with n flips the integral is always `1/(n+1)`.) So the curve is not normalised over the parameter, and there is no reason it should be — normalisation was imposed on the data axis, not on the parameter axis. Second, meaning. In the classical set-up `p` is a fixed unknown constant, not a random quantity, so there is nothing for a density over `p` to describe. A distribution over `p` only comes into existence when you supply a prior and multiply: the posterior is proportional to prior times likelihood, and the prior is the piece that makes the product integrable to 1 in a meaningful way. ### Only shape and ratios matter Because a likelihood is used through ratios, the constant `C(10,7) = 120` is decorative. Comparing two candidate biases, the factor cancels: `L(0.7) / L(0.5) = (0.7^7 * 0.3^3) / (0.5^10)` and the same cancellation happens inside Bayes rule, where any constant factor is swallowed by the normaliser. This is why textbooks say a likelihood is defined only up to a positive multiplicative constant, and why two experiments whose likelihoods are proportional carry the same information about the parameter. ### Reading the curve A likelihood curve has two features worth naming out loud in an interview. Its peak says which parameter value best accounts for the data. Its width says how sharply the data discriminate between values. With 7 heads in 10 flips the curve is broad: `p = 0.5` still has appreciable support, so the data are far from settling whether the coin is fair. With 70 heads in 100 flips the peak sits at the same 0.7 but the curve is far narrower, and 0.5 is effectively excluded. Same peak, very different evidence — a point a candidate who only quotes the peak will miss. ### Traps to avoid - Saying `L(0.7)` is the probability that the bias equals 0.7. It is not a probability of anything about `p`; it is the probability of the observed data under that value of `p`. - Swapping the conditioning: `P(data | p)` and `P(p | data)` are different objects, and only the second requires a prior. - Comparing likelihood values across different datasets. Within one dataset ratios are meaningful; across datasets the arbitrary constants differ and the comparison is empty.