Lenny Conundrum 277 and 278

[AySz88]

What percentage of users answering this question will get it correct?

Please round to the nearest whole number. Please do not include any characters in your answer other than numbers. This includes spaces, punctuation, or percentage signs. If there is no correct answer, we'll pick whatever answer comes closest to being correct.

For some ways to think about this problem, take a peek at the discussion when Slashdot asked "How many people will choose the same option as you?". Note that the bins were much larger in Slashdot's case, so this LC will have a much different result. Also note that if you guess a large number and guess correctly, you will get a smaller reward (since you'll be splitting the pot with more people).

I think the most important insight from there is:

This isn't a question of statistics. It's a question of psychology.

It's possible for there to be multiple correct answers, but I think this is rather unlikely. I would assume that they're using non-rounded percentages to determine the winner, so there's very little chance of having even one exactly-correct answer.

Update: Answer to LC 277:

3%. 4.1% got the conundrum correct, for an error of 1.1%.

Text of their analysis:
19,948 users entered this week's Lenny Conundrum. Here were the top ten answers, with the rounded percentage of users answering each one:

1. "100" (1370 guesses, 6.9% of total entries)
2. "1" (1141 guesses, 5.7% of total entries)
3. "3" (821 guesses, 4.1% of total entries)
4. "0" (766 guesses, 3.8% of total entries)
5. "2" (668 guesses, 3.4% of total entries)
6. "50" (654 guesses, 3.3% of total entries)
7. "5" (571 guesses, 2.8% of total entries)
8. "10" (525 guesses, 2.6% of total entries)
9. "4" (460 guesses, 2.3% of total entries)
10. "100%" (366 guesses, 1.8% of total entries, but DISQUALIFIED since the question stated to omit percentage signs)

All other guesses came in at under 2%, and since 0, 1, and 2 are already accounted for in the top ten, it's just a matter of which one comes closest. From above, you can see that "3" came closest to its corresponding percentage, so 3 was the correct answer. Wouldn't be interesting to see how people would answer this question already knowing how people answered it before?

And indeed, conundrum 278 is:

What percentage of users answering this question will get it correct?

Please round to the nearest whole number. Please do not include any characters in your answer other than numbers. This includes spaces, punctuation, or percentage signs. If there is no correct answer, we'll pick whatever answer comes closest to being correct. And yes, we're intentionally asking this question again.

[Siniri]

This is an awesome Lenny Conundrum. It is possible that there could be more than one correct answer, but if they're making us round to a whole number, then their answer must also be rounded to the nearest whole number. I'll be quite interested to see what the answer ends up being...

I made a guess towards the lower end of the spectrum; I would put my chances at about 1:20.

[AySz88]

I guessed too high... Oh well, one more try!

One way to get a handle on the first conundrum:
1) Not everyone is going to guess reasonably. Consider how many entries are definitely going to be wrong, such as disqualifications and "100" and other random guesses near the high end. 10%? 5%? 25%?
2) Out of the remainder, assume that reasonable people will randomly guess from what they consider other reasonable people will guess. This will likely mean that guesses will be randomly distributed across the lowest X percentage points. For which Xs is it reasonable that someone might be correct? (For example, 20% would be an unreasonably high guess: even if nobody was disqualified and people uniformly guess from 1 to 20, you'd only expect 5% for each choice. In other words, there wouldn't be enough people considering 20 for 20 to be the correct answer.)
3) The largest reasonable X is an upper bound on what you should guess.

Just one pass through this logic demonstrates that the correct answer will almost certainly be less than 10%. (10%: everyone guesses uniformly from zero to ten, with no disqualifications, resulting in a winning answer of 9%.) You can then go back to step one and ask yourself the slightly more concrete question "how many will be disqualified or guess above X?" and repeat.

The second iteration of this question is going to be a bit different: more people will start guessing numbers near 3, and you have to tweak your definition of "reasonable".

(I absolutely love these questions. *grins*)