Placing only 23 keys into a hash table with 365 entries with
    a perfectly random hash function will probably result in at
    least one collision.

No, it won't "probably" result in at least one collision, it will result in at least one collision with probability about 1/2. As quoted elsewhere, placing k objects in a hash table with N entries will result in no collisions with probability about exp(-(k^2)/(2N)).

Interesting. I was going to complain, as I thought "probably" meant the probability of occurrence was greater than 50%.. That doesn't seem to be the case.

  Probably: almost certainly; as far as one knows or can tell

I would guess waynecochran thought something similar.

Curious. I've just taken a straw poll in the office. I asked "if someone said something was probably going to happen, what probability would you give it?"

The answers were:

* 80%

* 50% to 90%

* 90%

* > 80%

* 80%

* 80% to 90%

My guess would have been "exactly strictly higher than 50%".

While mathematically correct, this does not seem to be how people use with that phrase. "X is probably going to happen" implies there is some evidence in favor of X, while "X is probably not going to happen" implies evidence against. Now, in regular speech, people do not feel compelled to force themselves to choose one or the other, but are just as likely to say "I have no idea", implying no evidence in either direction.

From an Information Theory point of view, 50% probability means zero information, and 25%/75% probabilities mean one bit of information against/in-favor-of the event. This sounds roughly consistent with the results of the poll, with people assigning higher probabilities implying higher information density of the evidence.

Same; "more probable than not", which falls back on your definition.

Yes, you're probably right.