From a standard (52-card, well-shuffled) deck of cards, the player is dealt seven cards, face down.

probability/card/guess.suit New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

From a standard (52-card, well-shuffled) deck of cards, the player is
dealt seven cards, face down.  The dealer then asks the player to guess
the suit (hearts, spades, clubs, diamonds) of the first card.  The
player chooses, and the dealer reveals the card.  If the player is
right, he gets a point.  If the player gets at least four points out of
the seven cards, he wins.

1. What is the player's best strategy?

2. What is the probability that the player will win using this
   strategy?

-- Scott M. Harker <sharker@saul.cis.upenn.edu>

Show solution

1. Assuming truly random distribution of suits within the deck, select
suits that haven't "shown up" yet.

For example, if the first card was a spade, the probability that the
second card will be a club (or a diamond, or a heart) is 13/51.  The
probability that the second card will be a spade is only 12/51.

2. Your probability of being correct on the first card is 1/4.

Using the "best" strategy:

Your probability of being correct on the second card is 13/51

Your probability of being correct on the third card is 13/50

Your probability of being correct on the fourth card is 13/49

Your probability of being correct on the 5th, 6th and 7th cards
depends on the suit distribution of the first 4 (or 5, or 6) cards.
The probability on the 5th card will be 13/48 or 12/48.  The
probability on the 6th card will be 13/47 or 12/47.  The probability on
the 7th card will be 13/46 or 12/46.

For convenience sake, our answer shouldn't be too far off if we just
use the approximation .265* as the probability for success on any one
of the seven cards and ask how likely it is that a player can "get a
point" at least four out of seven times.

There is one way to get seven points.  The probability of this is
approximately .265^7, which is approximately .0000917

There are seven ways to get six points.  The probability of this is
approximately 6(.735)(.265^6), which is approximately .001527

[ Editor: with seven ways the term is 7(.735)(.265^6) = .001782, which
  makes the total below about .0852 rather than .0850. ]

There are twenty-one ways to get five points.  The probability of
this is approximately 21(.735^2)(.265^5), which is approximately
.0148218

There are thirty-five ways to get four points.  The probability of
this is approximately 35(.735^3)(.265^4), which is approximately .06853

The combined probability of a player getting at least 4 points is
approximately .0849705.  In other words, a player will win
approximately 8.5% of the time.

*  Note:  if a player were to "pre-guess" all seven cards before any of
them were revealed, this number would be only .250.  I'll leave it to
the reader to see what chance of "winning" such non-strategy would
yield

-- Gary Scott Simon <garscosi@pipeline.com>