Question

$A$ draws two cards at random from a pack of $52$ cards. After returning them to the pack and shuffling it, $B$ draws two cards at random. The probability that their draws contain exactly one common card is

• A. $\frac{25}{546}$
• B. $\frac{50}{663}$
• C. $\frac{25}{663}$
• D. none of these

Answer 1

Answer: (B)

$\frac{50}{663}$

A can select 2 cards in $C_2^{52}$ ways. B can select on of these in $C_1^2$ ways and 1 of the remaining 50 in $C_1^{50}$ ways.
Hence, probability $=\frac{C_2^{52}\ast C_1^2\ast C_1^{50}}{C_2^{52}\ast C_2^{52}}=\frac{100}{1326}=\frac{50}{663}$