What is the 2/3 the Average Experiment and How Can You Help?

[mathematicsma]

Okay, the experiment is over. I thank everyone who participated, you all helped. I ended it about a week ago, but then finals caught up with me, and I didn't have a chance to follow up. So here it is:

The official results:

http://www.touromathclub.org/math-puzzles/2-3-the-average/23-average-outcome

A few points: obviously, it is impossible for 2/3 the average to be over 66.6, because even if you assume everyone else is supremely stupid and picks 100, the correct answer is still a bit under 2/3 that (under, because your own choice skews the number a bit).

That said, there were still some selections way up there. So I presume there are four categories of people playing this game:

1) Those who take a random stab. They get confused by the question, and just decide to choose a number and forget it, or they can't be bothered. (Or decide to do it for a joke). Included in this group are the people who misunderstand the question.

People in this group have an expected mean of 50. I'd assume anybody who picks a number like 68 is in this group. But it's just as possible that someone who chose 17 was randomly guessing, too.

2)Those who think about the problem very mathematically. They assume that they're playing against a bunch of people as smart as they are, and that everyone will think about the problem the same way. They choose zero.

3) Those who start trying to figure out what everyone else is doing. This group is further broken down by the degree they take it to. One person may figure that the mean will initiallly be 50, so 2/3 that is 33. Factor in a few people who realize this, and pick 33, so the mean is a bit lower. But another person will take it a step further, and another even further, etc.

4)Those who think about it for a little while, and realize that they have absolutely no way of figuring it out. So they do a google search. They find wikipedia says that in at least one case, it fell around 21. So they take a stab around there.

Anyway, I'm not sure I fully digested the results yet, I need to think about them a little more and see if there's anything noteable.

Comments, anyone?

 

[lisab]

Results of the experiment, for those of us who are visual .

edit - fixed x axis

 

[Redbelly98]

Hey, thanks for getting back to us mathematicsma.

I picked 0 -- using the logical reasoning that, if everybody chose 0, we could ALL be winners!

After it was too late, I realized two things
1. Some people are bound to pick nonzero numbers.
2. The instructions did not specify that you had to pick an integer. Just a number between 0 and 100.

If I had it to do over, I would have picked 0.001, or something like that, expecting an average just slightly over 0 -- in which case I would be the closest. Now that the results are in, I see I would still have been way off.

Thanks again, the experiment was fun and the results interesting.

 

[mathematicsma]

Redbelly: when I first heard the experiment, I immediately thought I should choose zero. Then, like you, I realized that if even one person doesn't think like me, and chooses another number, my results are off. Like you, until I did some research, I never imagined it would be so high. Note that at the http://museumofmoney.org/exhibitions/games/guessnumber2.htm" , the average is 23, so the correct guess is 15.3. So our experiment is pretty close to theirs, which presumably has a much larger sample space.

Lisab: thanks for the edited chart. It's much better than the one I have on my website, especially because it highlights the fact that the most chosen range is the centered around the correct choice in a much clearer way. (That sentence was clumsy. Sorry.)