The key to the whole problem. For the paradox to work, the host must KNOW that the door that he opens or the cup that he turns over is not the right answer. If the host doesn't know, then switching offers no benefit. Joe.
a) yes you should play, player's side; but only if: 1. you have the bank-roll to ride a string of losers and the will power to stay with it until you make your expected $5 per roll profit. (You should check the binomial distribution to see the chances of having a string of losers at $55 a pop which would wipe you out.) 2. you know the wheel has a 60:40 bias - I assume this is disclosed. Since many will drop out, disillusioned, after a string of losers, in practice it probably pays to be the house, even with this bias and payout. b) Yes - you could step in between the player and the house as long as the player's win amount is > ~$92 and the house pays out $100 for wins. c) You could in theory charge up to $60 per roll but who would play? No doubt there are other considerations I haven't thought of.
Coming back to Let's Make a Deal, does anyone remember the behaviourial tendency of contestents? Did they stay or switch? I remember watching that show as a kid, seem to recall most people stayed with their original choice.
I have no facts but I agree with you. I seem to recall that most contestants stayed with their choice. I also seem to remember that most wound up with the donkey. Joe.
Ok just did some research on this: One study showed just 13% of people switch. Further, most people incorrectly believe the odds to be 50-50 once Monty reveals one of the doors. However, even given the assumption of equiprobability, the overwhelming majority stick with their original choice. People explain this along the lines of that they'd feel worse if they switched and their original choice turned out to be correct than they would if they stuck with their choice and lost. And that, to me is quite fascinating. The fact once we make a choice we become emotionally attached to it regardless of it's rationality. Wow, for sure that can be seen in this thread and trading in general. The hostility of cognitive illusions discussed earlier: "In von Randowâs (1993) book about the Monty Hall problem, the German science journalist described how he shifted his interest from mathematical to psychological issues after he realized that switching is indeed better. He raised the following three questions: Why were so many people, even those who were highly educated, deceived? Why are so many of them still convinced of the wrong answer? Why are they so enraged? Similarly, Piattelli-Palmarini remarked that âno other statistical puzzle comes so close to fooling all the people all the time . . . . The phenomenon is particularly interesting precisely because of its specificity, its reproducibility, and its immunity to higher education.â He went on to say 'even Nobel physicists systematically give the wrong answer, and . . . insist on it, and are ready to berate in print those who propose the right answer.'" http://socrates.berkeley.edu/~fitelson/148/krauss.pdf
But the story says the host flips over an empty cup. It says nothing about whether the host knew that the cup would be empty before he flipped it. He could have just taken a random stab at which cup to flip over. And yet still the probabilities remain the same. It is better to switch. So having the host know where the ball is is not a necessary condition for the story to work.
Indeed!! Why are they so enraged? It does get funny after a while. I have another one for all of you deep thinkers. This one generated about 100 pages of discussion on another forum. Interestingly, in this case I also initially came to the wrong, incorrect conclusion. However, with the help of an open mind and clear logic I eventually came around to the right answer. Here goes: Imagine a jet plane with frictionless wheels sitting on a frictionless conveyor belt. As the plane tries to take off the conveyor belt moves in the opposite direction. Will the plane be able to take off or will the conveyor belt moving in the opposite direction keep the plane stationary ?
I am no quant or Ph.D. but isn't this common sense: Every cup has the probability of 1/3 in the beginning. When a cup is taken away, the prob for each cup remains the same unless-> it is clearly stated that the cup taken away does not have the coin meaning only one of the 2 remaining cups have it. Otherwise it could be the cup being taken away having the coin. There is no paradox here: when someone ask you this kind of question during an interview, just stand up, stretch out your middle finger and walk off.
It does matter because if the host does not know where the ball is and turns up the cup with the coin, there is no opportunity to switch. That the host knows where the coin is and will only turn up a cup without the coin is what creates the conditional probability and therefore the paradox. Joe.
If there is no friction, the moving conveyor belt will have no effect on the plane, so as the belt moves the plane stays right where it was to begin with. Because with friction, the belt starting to move has to have enough energy to move the plane as well. Without the static force of friction, the belt only requires the energy to move the belt and the plane is not effected. So, no, the belt moving in the opposite direction will not have any impact on the plane accelerating to take off.