I take both "A" and "B", and the resultant $10,000.

If Mr. Lowell "predicts" that you will select both boxes, then there is nothing in box B. He clearly stated that the winner will get the contents of the box or boxes regardless of what he predicts. He will, logically predict that you will pick both boxes; you have lost nothing in the choice, and box A results in a guaranteed $10,000, but you cannot pick it without also selecting box B.

Box B does not contain the million dollars. Because you will get the contents of the box or boxes regardless of what he picks, then even if you select the empty box B alone, you cannot receive the million dollar check. Lowell knows this, and is banking on the audience member being confused by the terms of the deal. In this case, Lowell is not "predicting" the outcome; he's simply hedging his bets.

Or, was I completely off the mark? (either way, a guaranteed $10g trumps a POSSIBLE $1.01 million)

Great riddle. You take both boxes. The key to the riddle is that the wizard/con artist is under no obligation to actually put $1 million under box B, so box B has an effective value of $0. The rest is a red hering but a good exercise in calculating equity.

David St. Hubbins

Thanks for a better elaboration than I could have made, David. I agree, it's a great riddle.

(and even if I'm wrong, I still take both boxes. I'm greedy like that).

i'm not taking sides yet, but let me play devil's advocate.

look at this from a purely statistical perspective: over ten years, lowell has a track record of correctly predicting an outcome nine of ten times. there is no real reason to suspect that his odds are about to change.

so, if you choose A and B, nine of ten times you will make 10,000, and one of ten you will make 1,010,000. given that you had ten opportunities, you should come out 1,100,000 richer.

if, instead, you choose only B, nine of ten times you will make 1,000,000, and one of ten times you will make 0. thus, given ten opportunities you should come out ahead 9,000,000.

or, to put this in the perspective of expectation (for you poker players out there), if you choose boxes A and B, your expectation is 110,000 per guess. if you choose only B, your expectation is 900,000 per guess.

hmm.

tbt

But, if he guesses incorrectly (1 time out of 10), and you DO choose only "B", but he predicted you would choose "A" AND" B, you get nothing.

If money were wagered on the guess, the paradox would be different. But, in actuality, because no money was wagered, you risk absolutely nothing. Therefore, both boxes is the ONLY reasonable choice.

Put it this way: You only get to pick ONCE.

okay.

but (and remember, im still not picking sides) look at it this way: suppose there are two boxes in front of you, labelled A and B. box A has 10,000 in it; box B has a 90% chance of having 1,000,000 in it, as determined by a computer program using the most sophisticated random number generator available, using prime number variables and so-on. of course, if you are unlucky enough to select B on the 1 of 10 times that 1,000,000 isnt there, you get nothing.

do you choose A or B? is it the same (or statistically close enough) question?

tbt

Well, you brought up an interesting point on that one (smirk).

When I was in college, I was coerced into taking PreCalc, even though it had nothing to do with my major (because that was the level I was at in math). Anyway, the gist of it was, precalc was 5 days a week, at 8:00 AM. Given my extremely full schedule plus the need to work full time, this made it nearly impossible to do my daily homework and still fulfill my other obligations (school and otherwise) to any degree of competency.

Now, the problem with this particular class is that the instructor randomly checked 1/4 of the class' homework each day. He assigned each student a number and used a random number generator to select the students he would check. If the homework was less than half done, there was zero credit for it. This factored in largely in the final grade.
And so, I had my boss stagger my weekday work schedule so that I was only working two days during the week (remember, full time employment was a financial necessity). I did my homework completely on the nights I wasn't working, but given that it took 2 to 3 hours to complete, was unable to complete it on the other two days a week. I figured, if I'm only coming up 25% of the time, I will be able to complete the homework the remainder of the time and be able to finish enough to keep up the needed "C" average to retain my scholarships.

Mathematics worked BADLY against me, and I came up 60% of the time (a classmate and I figured it out). I dropped the class at the latest date that I could drop it, and, suffice it to say, it played a significant role in my not continuing college (too many "real life" obligations for full time schooling to be an option).

So, I don't play the odds. 100% is a FAR better percentage than 90%.

that is an excellent point. i was wondering if this thread would get to the risk factor.

there is no doubt that risk affects the actual statistics. look at the lottery. first, it is very, very rare that the jackpot will ever get to a point where it makes statistical sense to purchase a $1 ticket. if you are going to choose a lump sum rather than the annuity, you get less than the actual jackpot; if someone else hits the jackpot at the same time, you will split the jackpot; and taxes take a terrible toll on the odds. but when you include the risk factor, things change. do those twenty nickels lying around your sofa cushions really matter to you? most likely (though not certainly) the value of that dollar is significantly less than the monetary dollar. meanwhile, 10 million dollars may affect your life the same way 100 million would. it changes the odds.

that was why i selected the figures for this equation that i did. for most people, 10,000 is a nice sum, one that you wouldnt want to throw away for a chance (albeit a great one) of taking down 1,000,000. without adding the risk factor, the question would be the same as putting 1 penny in box A and offering the chance at a dollar in box B. because of the risk factor, everyone would say the exact same thing: "duh, taboo. choose box B. who cares about a stupid penny?" not that you would care so much about a dollar, but a penny seems like nothing.

10,000 would be nice, though. theres a lot of things you could do with 10,000. still, as anyone who uses statistics to determine business decisions will tell you, all things being equal, you make the choice that works with the odds. it's not like you have to pay 10,000 dollars here. you start with 0 dollars and no matter what, you aren't going to lose. so unless you are in immediate jeopardy of dying or losing something important to you, this contest isnt going to sink you regardless of what happens.

Right you are on that regard.

However, assume that, on picking both boxes (remember, I still have a 10% chance of making $1.01 million...not the WORST odds in the world, actually), I return to my seat and the guy next to me comments that my decision was stupid, I'd have to reply "well, I got $10k...what did YOU get?" LOL

lol. nicely said.

i would guess that you do not have the gambling bug, and i commend you for it. there is a guy named "amarillo slim" preston, who, among other things, is a professional poker player. he also is known far and wide as the best proposition gambler alive. basically, he is golden when he establishes the odds of events playing out in certain ways. the rumor is that he has done everything from holding his breath in cold water to mass-twinkie consumption, when someone offered him "good odds".

or, to put it another way, he is a jerky nutjob. but he has become quite rich, too--though he has gone bust several times in the process. just because you pick the option with the best odds doesnt mean youre going to hit.

tbt

Exactly. And, as I said, I'm not entirely dismissing the 10% possibility that Lowell could guess wrong. That would result in an even further increase.

Ironically enough, I qualify as an investor with a high risk tolerance. Every profile I have ever taken has listed me as such. It took me awhile, though, to realize that "high risk tolerance" and "gambling" are two different things.

There are so many interesting thoughts that arise out of this, but I see it as basically a business decision, and part of what makes me a fiscal conservative. if the terms were changed, and I were to wager $9K on the same scenario, I'd choose only box "B", assuming I was assured that the game was on the level. The potential increase makes it well worth the assumed 10% risk. (THIS scenario is actually played out regularly, as when stockholders choose low to moderate risk stocks over bonds and CDs for investments).

that IS interesting, how the perspective shifts, huh?

you know, whenever i ask this riddle to friends, i get the OPPOSITE response, almost invariably! I wonder why? everyone says that you would be a fool to pick anything other than "B". 9 of 10 isnt a lock, but it's pretty good!

and that's when i usually launch into my free will discussion which hasnt even come now. the question i usually ask is whether or not there is any way on life that lowell can change his choice now. of course not. his choice has already been made. looking at it this way you have to choose a and b. the money is in there or it isnt, regardless of what you choose...or is it regardless of what you choose?

thats the real paradox of the question: can your choice really affect his decision?

tbt

TBT, I participate in certain game theory forums and I've seen variations of this puzzle before, so I'll stick with my first answer. To elaborate further, all pot odds type calculations for this puzzle are based on the faulty premise that Lowell will ever actually put the $1 million under a box.

Here is another variation for context: pick a hand (you pick left hand - or right, doesn't matter). I open my hand and have $10,000. Now in my other hand I tell you that I have either $1 million or $ one thousand, do you keep the $10K or go for the other hand? You keep the $10k, because you can't accurately ascertain what % of the time there is $1 million. If you knew for sure that 50% of the time there was $1 million in the other hand and the other 50% $1k then you can do a pot odds type calculation and that's where it gets interesting.

Everyone has a different indefference curve with respect to equity and efficiency so in a sense there is no right answer, even if one scenario had a 10X or greater EV.

For example, suppose you are offered either ten dollars, or a one in a million chance at $15 million. From an equity point of view the latter is worth fifteen bucks and is therefore better. But a heroin junkie would greatly prefer the ten bucks while Bill Gates might prefer the longshot overlay, and both can be considered correct due to their separate indifference curves. While in the long run you generally want the best of it equity wise, you need to take into account risk too.

David St. Hubbins

I take the empty box. Sure it might have lots of money in it, and that would be nice, but it would be so much more fun to prove him wrong live on national television. And to top it off I'd go down in history as the fool that picked an empty box over one I knew had money in it

about the risk factor, i completely agree. i would suggest, however, that this riddle is a little different from the "bird in the hand" riddle you proposed. as you correctly pointed out, there is no way to determine how often the two different sums will surface if you make the switch.

the laws of this riddle state that lowell has a track record of 90% successful prediction. lowell says that he has no concern as to whether people think he's a jerk (for not putting the money in , and he further suggests that his only interest is proving the validity of his track record for the public. of course, in real life we wouldnt take these statements as fact; nevertheless, while not out of the range of conceivability, a 90% successful prediction rate is unheard of. in the name of the riddle, we should assume that lowell will indeed put the money in box B if and only if he predicts that you will make "only B" your selection.

im glad you brought up game theory. if you have some gt knowledge, you are probably aware of the prisoner dilemna. without sidetracking into a new riddle, the relevant game theory principle is "defection". obviously, you stand to increase your "victory" by defecting (you will make 1,010,000 by "defecting" to a + b if he does not defect). if you both defect, however, you will gain a little, and he will gain the most (he only pays 10,000 and his track record is upheld). if you stay true by not defecting and he does defect, then you both will lose. from a game theory perspective, your best move is to not defect. in order for him to maintain any victory at all, lowell will have to meet you on your ground and give you the money . . . and, playing by the rules of the riddle, we should assume that paying out the 1,000,000 is fine with lowell, so long as his prediction is correct.

so, not including the element of risk, statistics and game theory both suggest that you should only select B. if you add the risk factor, the question produces, as you said, variable answers. but the real paradox still rests with the concept of free will.

tbt