# 407 probability/flush.p

## Description

This article is from the Puzzles FAQ,
by Chris Cole chris@questrel.questrel.com and Matthew Daly
mwdaly@pobox.com with numerous contributions by others.

# 407 probability/flush.p

Which set contains proportionately more flushes than the set of all

possible poker hands?

(1) Hands whose first card is an ace

(2) Hands whose first card is the ace of spades

(3) Hands with at least one ace

(4) Hands with the ace of spades

probability/flush.s

An arbitrary hand can have two aces but a flush hand can't. The

average number of aces that appear in flush hands is the same as the

average number of aces in arbitrary hands, but the aces are spread out

more evenly for the flush hands, so set #3 contains a higher fraction

of flushes.

Aces of spades, on the other hand, are spread out the same way over

possible hands as they are over flush hands, since there is only one of

them in the deck. Whether or not a hand is flush is based solely on a

comparison between different cards in the hand, so looking at just one

card is necessarily uninformative. So the other sets contain the same

fraction of flushes as the set of all possible hands.

Continue to: