Question

Four cards are drawn at a time from a pack of $52$ playing cards. Find the probability of getting all the $4$ cards of the same suit.

• A. $\frac{44}{4165}$
• B. $\frac{11}{4165}$
• C. $\frac{22}{4165}$
• D. none of these

Answer 1

The correct option is A $\frac{44}{4165}$

Since $4$ cards can be drawn at a time from a pack of $52$ cards is ${}^{52}{C}_4$ ways,
total number of elementary events ${=}^{52}{C}_4$
Consider the following events:
$A:$ Getting all spade cards;
$B:$ Getting all club cards;
$C:$ Getting all diamond cards;
$D:$ Getting all hearts cards.
Then $A,B,C$ and $D$ are mutually exclusive events such that
$P\left(A\right)=\frac{{}^{13}{C}_4}{{}^{52}{C}_4},P\left(B\right)=\frac{{}^{13}{C}_4}{{}^{52}{C}_4},P\left(C\right)=\frac{{}^{13}{C}_4}{{}^{52}{C}_4}$ and $P\left(D\right)=\frac{{}^{13}{C}_4}{{}^{52}{C}_4}$
Now required probability
$=P(A\cup B\cup C\cup D)$
$=P\left(A\right)+P\left(B\right)+P\left(C\right)+P\left(D\right)$
$=4\left(\frac{{}^{13}{C}_4}{{}^{52}{C}_4}\right)=\frac{44}{4165}$