From: dseal@armltd.co.uk (David Seal)
Newsgroups: rec.puzzles
Subject: Re: Another brain teaser
Date: 30 Jul 92 14:22:16 GMT
Distribution: rec
Organization: A.R.M. Ltd, Swaffham Bulbeck, Cambs, UK

In article <1992Jul28.210207.16460@athena.mit.edu> jimhsu@athena.mit.edu
(Jimmy M.D. Hsu) writes:

>	Brian is playing a game with his friend, Sam.  They use a complete poker
>deck with the two Jokers.  Sam shuffles the deck thoroughly and spreads all
>the cards out in one pile.  "This is pile A," says Sam.  "We take turns
>putting cards in pile B, which is the garbage pile and which is currently
>empty."  Sam points to an empty spot on the table.
>	"To get cards, you can take as many as you like from anywhere in pile A,
>but never from pile B," Sam continued.  "You then discard one of your cards
>into pile B and it's the next player's turn.  The object of the game is to
>avoid picking up the Jokers in pile A.  If anyone takes a Joker, that person
>loses and we start a new round. Whoever loses ten times is out of the game."
>	Brian decides to think about the game to determine the best strategy. 
>Since the probability is lower in the beginning and nearly certain towards
>the end when few cards remain in pile A, Brian decides to start his rounds
>by taking 27 cards, which is half of pile A.  Then, he reasoned, he will
>have gotten all the low probability cards. ...

This is the error in his reasoning. Assuming the shuffle to have been done
properly, all cards have equal probabilities of being jokers at the start of
the game. There's no reason why the jokers should have gravitated to any
particular place in the pack, and there is no such thing as a "low
probability card".

The reason the probabilities increase during the game is not that some cards
had a higher probability of being a joker from the start, but that the
players are acquiring extra information. When the first player plays a turn
successfully, taking N cards, both players learn that the first N cards
taken were *not* jokers: this is what increases the chance that the others
*are* jokers. (Similarly, an unsuccessful turn generally decreases the
chance that the others are jokers - but the players aren't really interested
in this, since the game is over at that point.)

>... He will then play his cards and force his opponent to take the Joker in
>the remaining cards.
>	Sam, being a good friend, points out that taking half the deck results in a
>75% chance of immediate loss, since there are two Jokers in the deck. ...

A nit-pick: the actual chance of immediate loss is slightly higher than
this. The chance that both Jokers are in the untouched half of the deck is
27/54 * 26/53 = 13/53, so the chance of immediate loss is 40/53, or about
75.47%.

>... He says his strategy is to take about six cards at the beginning, to
>build up a reservoir of cards.  He then uses all of his reservoir when
>around 13 cards have been played.

>	Is there a flaw in Brian's reasoning?  If so, where?  What is the best
>strategy?

Yes, there is a flaw in Brian's reasoning, as described above. In fact,
Sam's strategy suffers from the same flaw - i.e. he believes that there are
such things as "low probability cards" and that you need to stock up on them
at the start. In reality, all untaken cards have the same probability of
being jokers: taking more than you need to is simply an unnecessary risk.

The best strategy is always to pick up one card and immediately discard it
if you didn't lose by picking up a joker. Every time you pick up a card, you
take a chance of losing. Picking up more cards than you need immediately
simply increases your chances of losing before your opponent does.

E.g. compare picking up two cards now, then playing them off one by one,
with picking up one card now, playing it off, and repeating next turn. With
the first technique, I will lose if either card is a joker; with the second,
I will lose if either card is a joker, *unless* it's the second one I pick
up and my opponent picks up the other joker on his intervening turn. Picking
up both cards now forgoes this extra chance of winning.

>Variations to come...

Do you have to??? This is effectively a variation on the Monty Haul theme,
and that has been debated to death on this group...

To summarise the basic theme behind all such problems: the probability you
assess for any particular event depends on what information you have about
that event. You re-assess the probability when you get more information -
either positive or negative - about that event; you *don't* re-assess the
probability if the additional information you get isn't about that event.
Finally, the fact that the probability was re-assessed at some point does
*not* mean that the probability was equal to its re-assessed value from the
start: gaining the additional information really did *change* the
probability.

David Seal
dseal@armltd.co.uk

All opinions are mine only...
Newsgroups: rec.puzzles