Likelihood and Probability are usually thought of as synonyms, but in fact they have directly opposed meanings (if you are a statistician).
I often talk to my students about how to discuss likelihood and express probability and odds in English, and I usually tell them that likelihood and probability are synonyms, with likelihood being more conversational and probability perhaps more precisely calculated. If you ask someone about the probability of rain, it will sound like you want your answer expressed as a
percentage; if you ask about the likelihood, a shrug or a snort of derision may very well do.
But in statistical terms, likelihood and probability are opposites. That is, likelihood expresses the chance that a hypothesis is true given an observation or a series of observations; and probability expresses the opposite: the chance that an observable instance will occur given the truth of a hypothesis.
Whom do we have to thank for this admirable and sadly neglected increase in the precision of the language? R.A. Fisher.
Sir Ronald Aylmer Fisher (1980-1962), a graduate of Gonville and Caius College, Cambridge, was a pioneer in statistical biology, (he was, to be precise, a biostatistician) who ran a research institute at Rothamstead in Hertfordshire, just south of Cambridge, and established the field of the analysis of variance, working with crop data going back to the 1840s to unpick the variables causing variation in crop characteristics. His work was fundamental to the synthesis of Darwinian natural selection and Mendelian genetics. He has been called the ‘greatest biologist since Darwin’ (Richard Dawkins) and a genius responsible for ‘the foundations of modern statistical science' (Anders Hald).
But he is not, perhaps, so well know. Statistics, however vital, is in the popular imagination a dry pursuit, a dusty science. And yet our failure to think can often be traced to our failure to think statistically. In the context of likelihood and probability (loosely understood), I sometimes give my students the following brain-teaser (nicked from Daniel Kahnemann's Thinking, Fast and Slow):
You hear a young woman described as a “shy poetry lover”. Do you think she is more likely [probable?] to study Chinese literature or business administration?
The answer, as R.A. Fisher would have instantly seen, is business administration, since vastly more students study that than Chinese literature. Information about the young lady's interests is largely irrelevant. And there we have the beauty of R.A. Fisher and his rows of peas and beans – he was able to cut through the noise and mess of real life and isolate the salient variables. The question remains, however, whether allowing my students a glimpse of this clarity by more precisely glossing likelihood and probability for their benefit is in itself a move in the direction of clarity, or of noise. I rather fear the latter.