# Re: Can you help?

scot@flume.cs.dartmouth.edu
Tue, 29 Apr 1997 23:16:31 -0400 (EDT)

Duncan wrote:

> So, if I change the [brain] to [heart] in the problem:
>
> "> > > An alien picks a jar at random from a stockroom. The stockroom
> contains
> > > > an equal number of jars containing a single brain and jars containing a
> > > > single heart. The alien then adds a brain to the jar, shakes it, and
> > > > removes at random one of the two organs in the jar. The organ removed turns
> > > > out to be a [heart]. What is the probability that the original jar contained a
> > > > [heart]? "
>
> the probability then is 1? [Case 4) above.]
>
> I shall test this in our next poker game --- a sure win, right? Thanks.

Exactly. If the alien removes a heart from the jar, you know with 100%
certainty that the heart was in the jar originally, before he added the
brain. How else could it have gotten there? Therefore the odds that
the jar originally contained a heart is 1.

The poker equivalent is you are playing 5-card stud and only you and
one other player are left in the game. Between your up cards, your hole
card, and the cards of everyone who folded (they obligingly revealed
their hole cards) you have seen 51 cards of the 52 card deck, and you
have not seen the ace of spades. You know with probability 1 that your
opponent's hole card is the ace of spades. Originally you knew only
that there was a 1/52 chance that that card was the ace of spades, but
each card that you saw eliminated a possible hole card for your opponent,
and after eliminating 51 other possibiities the odds that that hole
card is the ace of spades is indeed 1.

This is an extreme example of what card counters in blackjack (Bro. Pat's
example) and poker players do all the time. I have seen a game of Anaconda
where the following happened. (Anaconda is a card game with lots of passing
of cards, where in the end each player picks 5 cards which are revealed one
at a time, with 5 rounds of betting, one after each card). One player's
first cards were the Ace, King, Queen, and Jack of spades. It looked as if
he could have a royal flush - the highest possible hand. If he did, his
last card would be the 10 of spades. An opponent had a full house or some
other hand that would win if the last card were not the 10 of spades. The
player with the apparent royal flush bet the limit on the first three rounds.
However, he dropped after the fourth round. One of the other opponents
turned up the 10 of spades, so the probability that it was a real royal
flush suddenly fell to 0!

Scot