Given our current knowledge of the world. We believe that the likelihood of a certain event is 50 percent. That is, we can’t really explain it very well, it just seems random.

Now say some new hypothesis comes along, call it the hypothesis A, this new hypothesis predicts that the event that we were curious about before actually occurs at a different rate. A says that if we condition on some parameters that it explicitly specifies then the probability that this event will happen is actually 90%.

Now let’s say that we take 5 observations and the event has only occurred once. The probability of this occurring with hypothesis A is p(1-p)^4=0.00009. Whilst the probability of it occurring with the original knowledge is 0.03125. So we should prefer the original hypothesis to this one.

In the example above, we used 5 observations to try and compare our two hypotheses. So what did we actually do? Well, we compared the difference between their probabilities and depending on whether the difference was positive or negative we chose one or the other. Now it should be easy to see that the larger is this difference, the more confident we can be in our rejection or corroboration (“temporarily accepting” in Popper’s language)of the new hypothesis.

So now assume that we have some other hypothesis, hypothesis B, and this hypothesis predicts that actually, this event will be occurring 50.1% of the time. Now it should be trivial to see the difference in probability between our current state of the world and this new hypothesis prediction will be very small indeed. In other words, this hypothesis will be much harder to falsify.

What this means is that the more likely something is given our current state of the world, the harder it is to falsify, and the harder it is to falsify, the less scientific it is. Once again, if a hypothesis is very likely given what we know, then this hypothesis NOT scientific. Indeed it is the opposite, the more unlikely it is, the more scientific it is. Why is this? Because we can test it.

Further reading:

https://plato.stanford.edu/entries/popper/

https://en.wikipedia.org/wiki/Falsifiability