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 ${}^{52}{C}_4$ ways.

$\therefore n\left(S\right){=}^{52}{C}_4$

Let A = event where the four cards drawn are red and B = event where the four cards drawn are black

Then, $n\left(A\right){=}^{26}{C}_4$ and n(B) = ${}^{26}{C}_4$

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

A and B are mutually exclusive events.

$P(A\cap B)=0$

By addition theorem, we have:

$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

$=\frac{{}^{26}{C}_4}{{}^{52}{C}_4}=0$

= $2\times \frac{46}{17\times 49}=\frac{92}{833}$

Hence, the probability that all the drawn cards are of the same colour is

$\frac{92}{833}$