Question

Two cards are drawn one by one without replacement from a pack of $52$ cards. The probability that the second card is of higher rank than the first card is
(rank in increasing order can be taken from ace to king)

• A. $\frac{1}{17}$
• B. $\frac{8}{17}$
• C. $\frac{1}{2}$
• D. $\frac{16}{17}$

Answer 1

The correct option is B $\frac{8}{17}$

For second card = $P_{samerank\left(S\right)}+P_{higher\left(H\right)}+P_{lower\left(L\right)}=1$
$P_S=\frac{{}^{13}{C}_1{\times }^4{C}_2}{{}^{52}{C}_2}=\frac{1}{17}$
( $\because$ selecting one rank out of $13$ and then selecting two cards from the same rank)
Now $P_H=P_L$ because there is equal possibility that either the first card has the higher rank and or the second card has higher rank.
So $2P_H+\frac{1}{17}=1$
$P_H=\frac{8}{17}$