We've already determined that the "average" is 75 tries, and it is extremely likely to not that many, and that's just from blindly guessing on all 15 questions.

I think that if you are going to make the statement that the average is 75 tries, you can't logically say that it is "extremely likely" not to be that many - otherwise the average would be lower.

Probability is a lot of fun, and in this case not too hard to calculate. (This is a phrase that should strike fear into the heart of anyone who says it, because it means the chance of an error in what follows goes up tremendously!)

The probability of getting all 15 questions wrong in 75

or fewer tries by guessing randomly is 64% (rounding to nearest integer). A few other exemplars:

Probability of winning in:

10 or fewer tries = 13%

20 or fewer tries = 24%

30 or fewer tries = 33%

40 or fewer tries = 42%

50 or fewer tries = 49%

On the other hand, the probability of

not winning after 100 tries is 26%; and not having won after even 200 tries is still 7%. Egads.

(Edit: I failed to notice WesleyCrusher's fine response before I played with a spreadsheet. His explanation shows in addition that adding modest knowledge quickly reduces your expected necessary number of plays.)