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Question
From a well shuffled deck of 52 cards, 4 cards are drawn at random. What is the probability that all the drawn cards are of the same colour.
Solution
Out of 52 cards, four cards can be randomly chosen in 52C4 ways.
∴ n(S) = 52C4
Let A = event where the four cards drawn are red
and B = event where the four cards drawn are black
Then, n(A) = 26C4 and n(B) = 26C4
i.e. P (A ∩ B) = 0
By addition theorem, we have:
P (A ∪ B) = P(A) + P (B) - P (A ∩ B)
= \[\frac{^{26}{}{C}_4}{^{52}{}{C}_4} + \frac{^{26}{}{C}_4}{6{52}{}{C}_4}\] - 0
=\[\frac{46}{17 \times 49} + \frac{46}{17 \times 49}\]
= \[2 \times \frac{46}{17 \times 49} = \frac{92}{833}\]
Hence, the probability that all the drawn cards are of the same colour is \[\frac{92}{833}\] .
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