# Can the probability of an event be an irrational number?

I am wondering whether it is possible to construct an experiment, where the probability of occurrence of an event comes out to be an irrational number.

• Sure, the event that a dart thrown uniformly randomly in a square falls in the disk inscribed in the square.
– Did
Apr 24, 2015 at 10:56
• This is an interesting mixture of theory and practice. The dart, in theory, hits a point with irrational coordinates a.s. However, the point of the dart is a huge something compared to the required resolution. My feeling is that the human mind gets lost in itself when creating theories that are, practically successful though, related to irrational numbers.
– zoli
Apr 24, 2015 at 11:03

A famous example is Buffon's needle. A needle is tossed randomly onto a horizontal plane ruled with parallel lines whose distance apart is the same as the length of the needle. The probability that the needle crosses a line is $2/\pi$.

It depends on how do you define experiment. Theoretically you can take an unfair coin where the probability of the tail is $\sqrt{2}/2$. Or one can construct a dice with the area of one of the faces being irrational (but this example is more complex to settle correctly).

This very much depends on whether or not you want a real life experiment and its evaluation in which case this is a physics question and depends a lot on what theory you exactly subscribe to. To me it seems that due to the fact that it seems that space is discretized it should not be possible.

On the other hand if you ignore the Heisenberg principle and assume infinite approximability then you can certainly get experiments with irrational probabilities. You can even get strictly transcendental probabilities (rational multiples of $\pi$).

• Even if you consider that space is discretized, that does not imply that every discrete unit is equiprobable. Apr 24, 2015 at 16:53
• @BrianTung You are right of course. It's possible that two different states could have both irrational probabilities. But in a universe with a fine amount of states and more importantly inherently finite precision it seems strange to talk about irrationals.
– DRF
Apr 24, 2015 at 17:50
• Again, I don't see that there is inherent finite precision. Even if there are a finite number of states, the principle of indifference does not necessarily apply. Apr 24, 2015 at 18:38