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# 2 cards are drawn at random

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Manager
Joined: 19 Aug 2010
Posts: 77
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Kudos [?]: 20 [0], given: 2

2 cards are drawn at random [#permalink]  19 Dec 2010, 09:02
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Two cards are drawn at random from a pack of 52 cards. What is the probability that either both are black or both are queens?

Is there a way to solve it without using the combinations formula?
Math Expert
Joined: 02 Sep 2009
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Followers: 5078

Kudos [?]: 57113 [1] , given: 8806

Re: 2 cards are drawn at random [#permalink]  19 Dec 2010, 09:30
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Expert's post
medanova wrote:
Two cards are drawn at random from a pack of 52 cards. What is the probability that either both are black or both are queens?

Is there a way to solve it without using the combinations formula?

Yes, there is.

OR probability:
If Events A and B are independent, the probability that either Event A OR Event B occurs is: $$P(A \ or \ B) = P(A) + P(B) - P(A \ and \ B)$$.

This is basically the same as 2 overlapping sets formula:
{total # of items in groups A or B} = {# of items in group A} + {# of items in group B} - {# of items in A and B}.

Note that if event are mutually exclusive then $$P(A \ and \ B)=0$$ and the formula simplifies to: $$P(A \ or \ B) = P(A) + P(B)$$.

Also note that when we say "A or B occurs" we include three possibilities:
A occurs and B does not occur;
B occurs and A does not occur;
Both A and B occur.

AND probability:
When two events are independent, the probability of both occurring is the product of the probabilities of the individual events: $$P(A \ and \ B) = P(A)*P(B)$$.

This is basically the same as Principle of Multiplication: if one event can occur in $$m$$ ways and a second can occur independently of the first in $$n$$ ways, then the two events can occur in $$mn$$ ways.

Back to the original question:
Two cards are drawn at random from a pack of 52 cards. What is the probability that either both are black or both are queens?

$$P(BB \ or QQ)=P(BB)+P(QQ)-P(BQ \ and \ BQ)=\frac{26}{52}*\frac{25}{51}+\frac{4}{52}*\frac{3}{51}-\frac{2}{52}*\frac{1}{51}$$ (note $$P(BQ \ and \ BQ)$$ means the probability that first card is black queen and the second card is also black queen).

Hope its clear.
_________________
Re: 2 cards are drawn at random   [#permalink] 19 Dec 2010, 09:30
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