Some general question about statistics

Question

Consider an experiment of picking a real number between 0 and 1. Let's say I picked 0.5. Two interpretations are possible:

1. Something extraordinary has happened, because the probability of picking 0.5 from the interval $[0,1]$ is zero.
2. Nothing extraordinary has happened, because the probability that the number you picked is between 0 and 1 is 1.

Both interpretations seem reasonable, but they have completely different conclusions. What is going on?

Answer 1

Ex-post-facto reasoning is always a source of confusion. A great example is a coin-flipping exercise in a large group (I know Richard Dawkins used this one in one of his lectures).

1. Everyone is given a fair coin and flips it
2. If it is Heads, they sit down
3. Repeat until only one person is left (if everyone gets heads it's a do-over)

This process is guaranteed to end with one person left. 100%. However, you can imagine the starting population to be enormous (say 1 million people).

Now, the last person can ask "Gee, what are the odds that I would have been the winner?"

Well, before we ran the experiment it would be one in a million. But after the fact its 100%.

Same for your 0.5 -- if you said (before drawing a value) "I'm going to get 0.5" you would be almost surely wrong. But we know a number must be returned.

Therefore, nothing extraordinary has happened, you picked a number and then asked a question "what are the odds?" afterward. You would have done the same with any number returned.