Question

Two cards are drawn from a well shuffled pack of 52 cards. Find the

probability that either both are black or both are kings.

Answer 1

Two cards are drawn from a well shuffled deck of cards.

$\therefore n\left(S\right){=}^{52}{C}_2$

A be the event of getting black cards

$n\left(A\right){=}^{26}{C}_2$ [$\because$ There are 26 black cards]

$\therefore P\left(A\right)=\frac{{}^{26}{C}_2}{{}^{52}{C}_2}$ $=\frac{26\times 25}{52\times 51}$

B be the event of getting both king cards

$\therefore P\left(B\right)=\frac{{}^2{C}_2}{{}^{52}{C}_2}$

$\frac{4\times 3}{52\times 51}$ [$\because$ There are 4 king cards]

Also, $P(A\cap B)=\frac{{}^2{C}_2}{{}^{52}{C}_2}$

=$\frac{2\times 1}{52\times 51}$ [$\because$ Two king are black also]

Now,

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

=$\frac{26\times 25}{52\times 51}+\frac{4\times 3}{52\times 51}-\frac{2}{52\times 51}$

= $\frac{660}{52\times 51}=\frac{55}{221}$