# How to interprete likelihood equal to 1

I'm trying to interprete the Example 1 from the wikipedia page: the likelihood function of a coin flip with a single parameter p expressing how likely a head will come up.

The likelihood is defined as "the plausibility of a parameter value of the statistical model assumed to describe the observed data, given specific observed data".

It's clear to me that the plausibility of a coin being fair (p = 0.5) flipping the coin twice and observing two heads (likelihood = 0.25) is lower than the plausibility of any other model that assumes a higher probability for the head to come up.

But how does one interprete a likelihood = 1 with p = 1, i.e. a model that assumes a head will ALWAYS come up? I'm confident it doesn't mean we are 100% sure that that's the correct model but I cannot really state why not.

Likelihoods express the probability, that the observed event (two heads up) occurs given some parameters ($$p = 1$$). If you have a coin that always lands on head, two heads in a row have a probability of 1. However, the likelihood does not equal the probability, that these parameters / models are the correct ones. It is crucial to distinguish between plausibility and probability here.
• If likelihood is not equal to probability then why do we write this $L(\theta|O) = P(O|\theta)$? Doesnt $P(O|\theta)$ integrate to 1? – GENIVI-LEARNER Jan 23 at 13:30