Question

From a well shuffled pack of $52$ cards four card are drawn at random. find the probability that all are of the same suit.

Answer 1

Total number of cards $=52$

No. of cards for each suit $=13$

Total types of four suit $=4$ (heart, spades, diamonds, clubs)

Probability$=\frac{conditionalcase}{totalcase}$

For this conditional case of selecting all four same suit, it can be either heart or spades or clubs or diamonds.

So, total number of conditional case$=$No. of case of selecting all $4$ cards of hearts suit $+$ no. of case of selecting all $4$ cards of spades suit $+$ no. of case of selecting all $4$ cards of clubs suit $+$ no. of case of selecting all $4$ cards of diamonds suit

$={}^{13}{C}_4+{}^{13}{C}_4+{}^{13}{C}_4+{}^{13}{C}_4$

$=4\times {}^{13}{C}_4=\frac{4\times 13!}{9!4!}$

$=\frac{4\times 13\times 12\times 11\times 10\times 9!}{9!\times 4\times 3\times 2\times 1}=2860$

Total number of cases $={}^{52}{C}_4=\frac{52!}{48!4!}=270725$

So, probability of getting all $4$ cards of the same suit $=\frac{2860}{270725}=0.0106$.