Two cards are drawn from a well shuffled pack of 52 cards. Find the probability that either both are black or both are kings.
As 2 cards are drawn from a deck of 52 cards. This can be done in 52C2 ways. If S represents the sample space,
n(S) = 52C2
Let B represents the event that both drawn cards are black.
∵ A deck of 52 cards has 26 black cards. So 2 cards can be selected out of those 26 in 26C2 ways
∴ n(B) = 26C2
∴ P(B) =
Let K represents the event that both drawn cards are king.
∵ A deck of 52 cards has 4 king cards. So 2 cards can be selected out of those 4 in 4C2 ways
∴ n(K) = 4C2
∴ P(K) =
We need to find the probability that either both are black or both are kings i.e. P(B or K) = P(B ∪ K)
Note: By definition of P(A or B) under axiomatic approach(also called addition theorem) we know that:
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
∴ P(B ∪ K) = P(B) + P(K) – P(B ∩ K)
As we don’t have value of P(B ∩ K) so we will find it first.
As there is a common element among the events B and K as both the cards can be a king and can be black 2.
∵ 2 black king cards are present so we need to select 2 cards oout of them only. This can be done in 2C2 ways = 1
∴ P(B ∩ K) =
∴ P(B ∪ K) =