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Manager  Joined: 19 Aug 2010
Posts: 61
2 cards are drawn at random  [#permalink]

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Question Stats: 60% (01:12) correct 40% (01:29) wrong based on 31 sessions

### HideShow timer Statistics 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 V
Joined: 02 Sep 2009
Posts: 56257
Re: 2 cards are drawn at random  [#permalink]

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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.
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Re: 2 cards are drawn at random  [#permalink]

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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?

Probability that both are black = 26/52*25/51
Probability that both are Queens = 4/52*3/51
Case that is counted in both cases calculated above = (2/52)*(1/51)

Final Probability = (26*25+4*3-2*1)/(52*51)
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Re: 2 cards are drawn at random  [#permalink]

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See the solution.
refer to pic attached for the solution
Attachments IMG_1320.JPG [ 1.05 MiB | Viewed 11160 times ]

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Intern  Joined: 18 May 2018
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Re: 2 cards are drawn at random  [#permalink]

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How is it 2C2, can you please explain this?
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Re: 2 cards are drawn at random  [#permalink]

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_________________ Re: 2 cards are drawn at random   [#permalink] 27 Jun 2019, 01:17
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