Given a random variable $X$ which arise from a parameterized distribution $F(X;θ)$, the likelihood is defined as proportional to the probability of observed data as a function of $θ$: $\operatorname{L}(θ | x)=\operatorname{P}(X=x \mid θ)$
In statistics, a likelihood is a function of the parameters of a statistical model evaluated for a fixed sample of observations, defined as follows:
Likelihood — The likelihood that any parameter (or set of parameters) should have any assigned value (or set of values) is proportional to the probability that if this were so, the totality of observations should be that observed.
Fisher, Ronald A. "On the mathematical foundations of theoretical statistics." Philosophical transactions of the Royal Society of London. Series A, containing papers of a mathematical or physical character 222.594-604 (1922): 309-368.
Likelihood functions play a key role in statistical inference, especially methods of estimating a parameter using a statistic (a function of the data).
Excerpt reference: @ars's answer on What is the difference between “likelihood” and “probability”?
