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## Probabilities in physics.

Many people get spooked by the fact that in physics, especially in quantum mechanics, the only predictions that we can make are probabilistic in nature. Let us say you have a polarized photon, and you ask if it will pass through by another polarizer or not. The only answer you get is it will pass 60 percent of the time and that it will not pass 40 percent of the time. Moreover you can not tell in advance in which way it will go.

This is a fundamental fact about the world we live in. That fact does not mean that physics is unpredictive. Quite the contrary, you can predict those probabilities with remarkable precision.

However, this also means that making precise measurements in physics can require a lot of statistical manipulation, because changing the underlying physical parameters can only change the probability distributions for events, and even though some events might become more unlikely, they are not impossible. It might be instead that you got a freak observation (unlikely events do happen all the time).

So I though I would give you a taste for how to make predictions by giving you a statistical puzzle.

Let us say that someone gives you a lopsided bet. Such that with probability $r<1$ one gets heads, and with probability $(1-r)$ one gets tails, and you have to pick heads or tails.

You only know the outcome of the first event. After the first toss it came out heads let us say.

Now for the question I ask of you. What is the probability that $r>1/2$ ?

Excuse me?-You might ask- I thought the parameter r was a given.

And my answer is that, yes, someone else knows r, but he won’t reveal the answer, because if he does so he could lose a lot of money, so you have to make assumptions about how r is distributed to be able to answer this question in a meaningful way. Seeing as you have no information about r, you might as well assume that any value of r is equally likely and then the question above makes perfect sense.  Go have fun on this one.

Incidentally, similar problems do show up in physics all of the time. Let us say that the parameter r gets replaced by an angle of a polarizer (as above) and that your job is to find the best estimate for that angle with the knowledge of only one (or 5 or 15) polarized photons… If you ask astrophysics folks about polarization of light of very distant stars every single photon counts.