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Randomly Selecting a Card

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Date: 02/20/98 at 14:49:29
From: Sandra Jordan
Subject: probability

A card is drawn at random from a standard deck of 52 playing cards.
Find the probability that the card is a king and a club.

Here's how I have tried:

I know there is 13/52 clubs and 4/52 kings, and 1 of the kings shares
the club, so that would make 12/52 clubs, and 3/52 kings, and 1/52
king of clubs.

What do I do from here?
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Date: 02/20/98 at 16:17:08
From: Doctor Sam
Subject: Re: probability

Sandra,

I think you're making the problem harder than it needs to be.  There
is only one king of clubs in an ordinary deck.  So the probability of
picking it is 1/52.

A more difficult problem (which you are well on your way to solving)
is to find the probability that the card you pick is either a king OR
a club.  Now the probability is 16/52.

You can get this answer the hard way, by listing all possibilities:

the clubs:  A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, and
the kings:  of hearts, of spades, of diamonds

There are sixteen possibilities, so this answer is 16/52.

But there might have been too many possibilities to list.  Your method
can be used to answer this question.  Many people get this question
WRONG by assuming that

P(king OR club) = P(king) + P(club)
=  4/52   + 13/52.

But this counts the king of clubs twice: once as a king and once as a
club. The correct method is to subtract the extra time this counts the
king of clubs:

P(king OR club) = P(king) + P(club) - P(king AND club)
=  4/52   +  13/52  - 1/52
= 16/52.

I hope that helps.

-Doctor Sam,  The Math Forum
Check out our web site http://mathforum.org/dr.math/
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Associated Topics:
High School Probability
Middle School Probability

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