Can the probability of an event ever be a transcendental number?

Yes it can.

 

I don't think I can really answer your question unless you can explain why you think it would have to be a rational number.


For example, in a case where we have two (atomic) events A and B, there is absolutely no reason why we can't build a probability measure from the assignments:

P(A) = 1/pi,
P(B) = 1 - 1/pi.

 

In general,prob.s are not rational no.s.
If you can use the classical definition (due to Laplace) for some case then you get prob.s as rational no.s.

If you take Kolmogorov's axiomatic approach or in general think of prob. as a measure then they are not necessarily rational.

 

There are several definitions of probability, some only work intuitively. Personally I use the axiomatic approach, and measure theory. But, I hope that the following gives an intuition for how to get from one to the other.

If you are used to a probability being favorable/total outcomes, then imagine the idea of tossing a 6 sided die an infinite number of times. After one toss the probabilities are of the form n/6, after 2 tosses, n/36, after 3 tosses n/216, and so on: in the limit as the die is tossed an infinite number of times we have probabilities that are limits of sequences of rational numbers, which is what an irrational (cauchy sequences) numbers can be defined to be.

The axiomatic approach just says that we START with some probabilities, which could be ANY real number between 0 and 1, without trying to work out any combinatorial method of getting at these numbers.

 

It is true that P(A) can be thought of (always) as m(events that result in A)/m(total number of events), where m is a measure. Your mistake is in thinking that all probability measures come from events, and state spaces, with a finite number of elements, and that the measure is just counting the number of elements in each set.

In general neither of these is necessarily true.

If that sounds complicated, let's just think of an example. Let's take tossing a coin twice. What is the probability of getting at least one head. We can think of the sample space as

{HH,HT,TH,TT}

each of which is equally likely, thus we are in a situation where your intuiton works - the measure* of any singleton in the set is just 1, so we can count and the answer is 3/4.

But instead we could use the state space

{"string with no Hs", "string with exactly 1 H", string with exactly 2 Hs"}

Now we have to assign different measures to the elements in there. And in general there is no reason why we should have to assign integer measures and get rational probabilities.

Let's take the r.v. X which is uniformly chosen at random from [0,10]. What is the probability that X is in [0,pi]? We assign the measure of pi to the event in [0,pi], and the measure of 10 to the interval [0,10], and we see that the answer is pi/10.


* I'm not using measure here to mean probability measure.

 

Suppose a point p may lie any where on the interval [0, 10] with uniform probability. What is the probability that p lies on the interval [itex][0,\pi ][/itex]?

 

You are limiting your understanding of probability to a finite number of equally-likely outcomes -- drawing a card, flipping a coin, rolling dice. This is the classical definition of probability developed by Laplace. The concepts of probability and statistics have long since been extended to cover far more than gambling. For example, what is the probability it will rain tomorrow, or that the light bulb you just replaced will fail within a week? Classical probability theory cannot answer these questions, as the spaces are not enumerable. Modern probability theory can answer these kinds of questions.

 

MeJennifer: Can the probability of an event ever be a transcendental number?

There is a famous math problem from 1777, not long after Laplace in 1771 begain submitting papers on probability; this problem does fit the bill: Buffon's Needle Problem.

Here, we draw parallel lines on the floor, d apart, and drop a needle of length l on the floor. How likely is it that the needle will land on a line? For l=d, the needle being as long as the distance between the lines, the answer is 2/pi =.6366

http://mathworld.wolfram.com/BuffonsNeedleProblem.html