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Two cards are drawn from a pack of 52 cards. What is the probability that, both the cards are of same colour? - Mathematics and Statistics

Sum

Two cards are drawn from a pack of 52 cards. What is the probability that, both the cards are of same colour?

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Solution

Two cards can be drawn from 52 cards in 52C2 ways.
∴ n(S) = `""^52"C"_2`
Also, the pack of 52 cards consists of 26 red and 26 black cards.
Let A be the event that both cards are red.
∴ 2 red cards can be drawn in 26C2 ways.
∴ n(A) = `""^26"C"_2`

∴ P(A) = `("n"("A"))/("n"("S")) = (""^26"C"_2)/(""^52"C"_2)=(26xx25)/(52xx51)=25/102`
Let B be the event that both cards are black.
∴ 2 black cards can be drawn in 26C2 ways
∴ n(B) = `""^26"C"_2`
∴ P(B) = `("n"("B"))/("n"("S"))=(""^26"C"_2)/(""^52"C"_2)=(26xx25)/(52xx51)=25/102`
Since A and B are mutually exclusive and exhaustive events
∴ P(A ∩ B) = 0
∴ Required probability = P(A ∪ B)
∴ P(A ∪ B) = P(A) + P(B)

= `25/102 + 25/102 = 50/102`

P(A ∪ B) = `25/51`

Concept: Addition Theorem of Probability
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