Question

Two cards are randomly chosen one after the other from a standard deck of $52$ playing cards without replacement. What is the probability of choosing two kings?

• A. $\frac{1}{169}$
• B. $\frac{4}{663}$
• C. $\frac{1}{221}$
• D. $\frac{3}{676}$

Answer 1

The correct option is C $\frac{1}{221}$

Total number of cards in a deck= $52$
Number of kings in a deck= $4$

For choosing two kings, the first as well as the second picks should be king.

Suppose, the probabilities of choosing first king and second king are $\text{P(K1)}$ and $\text{P(K2)}$, respectively.

Now,
$\text{P(K1)}=\frac{\text{Number of kings in a deck}}{\text{Total number of cards in a deck}}$

$\Rightarrow \text{P(K1)}=\frac{4}{52}=\frac{1}{13}$

After choosing the first king without replacement,
total number of cards in the deck= $52-1=51$
number of kings in the deck = $4-1=3$

Now,
$\text{P(K2)}=\frac{\text{Number of kings in the current deck}}{\text{Total number of cards in the current deck}}$

$\Rightarrow \text{P(K2)}=\frac{3}{51}=\frac{1}{17}$

Finally, the probability of both the chosen cards to be kings,

$\text{P(K1 and K2)= P(K1)}\times \text{P(K2)}$
$\Rightarrow \text{P(K1 and K2)}=\frac{1}{13}\times \frac{1}{17}$
$\Rightarrow \text{P(K1 and K2)}=\frac{1\times 1}{13\times 17}=\frac{1}{221}$

$\therefore$ The probability of choosing both king cards without replacement is $\frac{1}{221}$.