Re: Probability question regarding Deal or no Deal

"John D'Errico" <woodchips@xxxxxxxxxxxxxxxx> wrote in
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"Richard " <rwkopcke@xxxxxxxxx> wrote in message
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waggonra <waggonra@xxxxxxxxxxx> wrote in message

<25760031.1204312690798.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>...
Am I the only one who doesn't think that the probability
of having the $1,000,000 case when there are only two cases
left, the one first chosen and the only one not chosen, is
not .5?



This is very similar to the Monty Hall problem. If I choose

No. You either misunderstand the Monty Hall
problem, or this one.


one from three doors, my prob of choosing the winning door
is 1/3. The prob of one of the two remaining doors holding
the prize is 2/3. Now suppose Monty shows me that one of the
two other doors is empty and gives me a chance to swap the
door I originally chose for the other of the two doors. I
should switch from my original choice (prob 1/3) to this
other door (prob 2/3, conditional on knowing that the second
of the two is empty). The probs would not change if Monty
had chosen one of the two remaining doors randomly and
happened to reveal that it was empty. This would only happen
about half of the time, though.

This is your misunderstanding.

Monty does not open one of the other doors
randomly. He ALWAYS opens a door that has
junk behind it. Only if both doors have junk
behind them is he unconstrained in his choice.

If Monty opened one of the other doors
randomly, and if the opened door had the
big prize or junk randomly behind it, then
the game changes, at least it does if you
know that he has opened a random door.


So for Deal or No Deal, assuming 26 cases: The prob of my
picking the 1m case at the start is 1/26. The prob of the 1m
case remaining in play is 25/26.

Let us imagine that we have chosen the
order of our picks from the very start, but
that the cases have been randomly assigned.
We can make this assumption without loss
of generality.

In effect, we will pick some ordering of the
case numbers 1-26, at the beginning of the
game. With probablility 1/26, case 1 (which
remains hidden) contains the big prize. Also
with the same probability, case #26 holds
that prize. Nothing about the further play of
the game changes these odds, since it is YOU
who has chosen the order. The TV host has
no impact on your choices. And if we assume
that the cases are indeed filled randomly at
the start, then the the odds of either the
first or the last case being correct are equal.
Those odds stay equal down to the very end.
Of course, if at some point in the game the
big prize is revealed, then the question is a
moot one.

The TV host only plays on your mind, trying
to keep the suspense up in a rather trivial
game. (I like the subtitle "Lotto for Dummies"
here.)

As Steve points out, an interesting aspect
of this game is in the payoff at any point in
time. You can compute the expected value
of the remaining cases easily enough. I'd
point out that if the payoff was always an
accurate measure of the true expected value
at any point in the game, then people will be
tempted to leave too early, too often. So the
trick is to ramp up the payoff towards the
true expected value as the game progresses.

One thing I've never watched the show long
enough to know, is if you go all the way to
the last case, do they give you an accurate
expected value pay off at the very end?

These payoff questions are of interest not
because of any issues of probability, as the
final expected value of the game at any point
is trivial to compute, at least if it is always
played to the very end. Of some interest is
the question of the maximum expected
value of the game, or even the maximum
expected payoff value. This would be of
interest to the game producers, since they
want to know what it will cost them on
average to produce this game. (I'd imagine
they might have an insurance company
policy on this, to even out the costs.)

The strategy issue is of interest. As the
game producer, you wish to entice the
player to stay an long as possible. You
want to keep the suspense up - long games
of this sort are far better for your ratings,
whereas games that end immediately are
not good at all. This is really psychology
though, not so much probability.

As the player, you would want to know
the approximate expected value of the
game at any point. This tells you when the
payoff value of the game is reasonable for
your own goals. Of course, all individuals
have different risk tolerances - this is one
of the things that make some game shows
mildly interesting.

John


I watched deal or no deal to nearly the end three times. The
offer starts out well below the expected value, but as the
number of cases drops, the offer approaches the expected
value. I have never seen a two-case ending, because the
contestant usually takes the offer once it is sufficiently
close to the expected value.
.

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