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}{{}^{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 $\frac{92}{833}$.