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.  
 A: 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 $2/\pi$.
A: 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).
A: 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 $\pi$). 
