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.

Answer 1

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

\[\Rightarrow P\left( A \right) = \frac{^{26}{}{C}_4}{^{52}{}{C}_4}\]  and

\[P\left( B \right) = \frac{^{26}{}{C}_4}{^{52}{}{C}_4}\]

A and B are mutually exclusive events.
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 \[\frac{92}{833}\] .