## Tuesday, May 19, 2009

### Puzzle

Here's a puzzle.

Suppose I made you an offer. I have two check made out to you. One check is hidden in my left hand and one check is hidden in my right hand. One of the checks is for an arbitrary, positive, non-zero amount and the other check is for an amount either ten times larger than the other check or one-tenth the amount of the other check. The amounts have arbitrary precision (i.e., the amounts can contain fractions of a penny). It is random which hand contains the larger check.

Here's the deal.

You can pick the check in either hand. After you look at the check in the hand you picked (but you're not allowed to see the other check), you have to decide which check you want: the check that you've seen in the hand you've picked or the other check that you haven't seen. The game is then over and you walk away with your check.

Here's the puzzle.

Common sense would tell you that since the contents of my hands are random and unknown, it shouldn't matter what you do. Pick the check in either hand and be done with it, or pick either hand and then choose the check in the second hand - shouldn't matter.

Common sense would be wrong.

Let's say you've chosen a hand and find that the check is for \$10. The other check is therefore either \$100 or \$1. The expected value is (\$100 + \$1) / 2 = \$50.50 which is substantially better than the \$10 you hold. So you should always go for the second hand after looking at the check in the first.

Which, of course, makes no sense since it's random.

What's the explanation?

The answer to this question is also the key to the secrets of the universe.

Bret said...

My co-blogger Howard knows the answer so I would prefer that he not comment on this particular post.

Hey Skipper said...

Reminds me of Jeopardy.

Bret said...

Jeopardy? Or do you mean The Price is Right?

In either case, this is a different puzzle.

erp said...

Mathematics is the key to the secrets of the universe, but I don't know how that explains the current dilemma unless it's that it's in man's nature to seek the unknown.

Don't keep us in suspense too long please.

Bret said...

I'll post an answer next week.

I've also given this puzzle to some co-workers and they haven't gotten it yet either. Some of them read this blog and are still intrigued with figuring out the puzzle so I can't post the answer till they get it or give up.

erp said...

Can you explain why each random pick between these your two checks isn't a 50:50 chance the same way that even flipping a coin endlessly, your chances are still 50:50 in picking heads each time you toss.

Peter Burnet said...

If you didn't look at the first check, then the expected value would be the mid-point between the two actual checks, no? You don't know what that number is, but you can describe it with reference to two existing checks and quantify it after the experiment. But by looking at the first check, your expected value suddenly becomes the mid-point between two possibilities, only one of which can be represented by an actual cheque. So your expected value changes, even though the odds of choosing the larger check remain 50-50 and your chances don't change.

Am I on to something or should I just go back to flailing away at Dawkins?

Bret said...

Peter,

Very close, but not quite.

The expected value is not the mid point between the two checks.

If you more carefully consider what the expected value of the initial problem is, then you could explain why it "suddenly" seems to become something after the observation of the first check.

Peter Burnet said...

Bret, you have no idea how out of my depth I am here, so be merciful. I just checked out expected value on Wikipedia and didn't understand a thing. however, I am assuming expected value is an objective statistical concept and has nothing to do with what is subjectively expected. If I am wrong, then it is over to AOG.

Using your example and method of calculation, if the checks are \$1 and \$10, then the expected value would be \$5.50 and if they were \$10 and \$100, it would be \$55.00. You don't know that, but it doesn't matter because they are what they are based on objective reality. But the moment you see the \$10 check, you calculate a 3rd different expected value that cannot be correct because the two cheques cannot be \$1 and \$100. If one exists, the other doesn't so it is a distortion to talk of probability as the odds of choosing the higher one. So it is the observation that distorts the underlying objective reality.

Come to think of it, it's over to AOG anyway.

Bret said...

Peter,

I'm just impressed you gave it a shot and you were on the right track in your first attempt.

By the way, "mean" or "average" are more or less the same thing as "expected value" for many problems. In this case, if we played this game a gazillion times, it would be the average amount of all the checks you received.

Susan's Husband said...

I thought about it a bit, but didn't come up with any interesting approach. I would say that the expected value before choosing is undefined (or possibly infinite).

Intuitively, you should just flip a coin and not consider switching, as you just as likely to have picked the larger check as the smaller, regardless of what the value you find in the first hand. I don't think the expected value is a useful metric in this case, as there is not enough information to make it well determined.

Bret said...

Susan's Husband,

You've identified the answer - the expectation is undefined (Not a Number or NaN for us programming nerds), allowing things to seem inconsistent when they really aren't.

To work backwards, in this particular problem, we know that the expected value (call it E) must be the same for either hand and yet the expected value for the 2nd hand is 5.05 times E (in other words 5.05 times the amount in the first hand). Therefore, we have

E = 5.05 E

The above holds (is not false) in three cases: when E is 0, infinite (∞), or undefined (NaN).

The game uses a uniform distribution on the interval (0, ∞). In otherwords, all check amounts are equally likely. The expected value is then (more or less not worrying about using limit notation) the integral of 1 / ∞ over that interval (0, ∞) which is essentially ∞ / ∞ and the result of that operation is NaN (undefined).

Thus, there is no mathematical inconsistency, because the equation

NaN = 5.05 NaN

is not false.

The reason I wrote that "the answer to this question is also the key to the secrets of the universe" is because the probability that the universe is in this exact state right now is undefined until you open your eyes and look at it. Just like the expected value of the check in the 2nd hand is undefined until you look at the check in the first hand.

Peter Burnet said...

The game uses a uniform distribution on the interval (0, ∞). In otherwords, all check amounts are equally likely. The expected value is then (more or less not worrying about using limit notation) the integral of 1 / ∞ over that interval (0, ∞) which is essentially ∞ / ∞ and the result of that operation is NaN (undefined).

Thus, there is no mathematical inconsistency, because the equation

NaN = 5.05 NaN
Surely it was clear that is what I meant, Bret. What was your problem? :-)

Someday, somewhere, we must do a thread on how wondrous the wonders of science truly can be to us little people.

Bret said...

Peter,

This was a math puzzle, not a science puzzle.

It's a nice example of how observation can "suddenly" change the system. The "Collapse" post at thought mesh reminded me of it.

Hey Skipper said...

Jeopardy? Or do you mean The Price is Right?

In either case, this is a different puzzle.
No, I don't think so.

Finding out what is behind one of the doors in Jeopardy gives you some information as to what is behind the other two.

Intuitively, it shouldn't. But it does.

Hey Skipper said...

Sorry, information is the wrong word.

It gives an expectation.

Susan's Husband said...

No, the door opening provides information about what is behind the doors because Monty knows where the prize is, and he shares some of that knowledge with you when he opens the door (because he always opens a non-prize door and you know that).

Bret said...

Hey Skipper,

In this case you don't actually need to even know the contents of the 1st hand to know that the expectation of the second is both the same as the first and 5.05 times the first at the same time. So it's not the same.

Isn't jeopardy the one where they show you an answer and you have to come up with the question?

Hey Skipper said...

Ooops.

I meant Let's Make a Deal -- it is called the Monty Hall problem.

The chances of winning increase if you change your choice after one of the other doors is opened.