Question

Four cards are successively drawn without replacement from a deck of $52$ playing cards. What is the probability that all the four cards are kings ?

• A. $\frac{1}{270725}$
• B. $\frac{1}{270125}$
• C. $\frac{1}{27025}$
• D. None of these

Answer 1

The correct option is A $\frac{1}{270725}$

Let $k$ denote the event that the card drawn is king.

now, $P\left(k\right)=\frac{4}{52}$

$P\left(\frac{k}{k_1}\right)$ is the probability of a second king with the condition that one king has already been drawn. Now there are $3$ kings in $(52-1)=51$cards

$P\left(\frac{k}{k_1}\right)=\frac{3}{51}$

$P\left(\frac{k}{k_2}\right)$ is the probability of a third king with the condition that two kings have already been drawn.Now there are $2$ kings in $(52-2)=50$cards

$P\left(\frac{k}{k_2}\right)=\frac{2}{50}$

$P\left(\frac{k}{k_3}\right)$ is the probability that three kings have already been drawn.Now there are $2$ kings in $(52-3)=49$cards

$P\left(\frac{k}{k_3}\right)=\frac{1}{49}$

Probability that all $4$ cards are king

$=P\left(k\right)\times P\left(\frac{k}{k_1}\right)\times P\left(\frac{k}{k_2}\right)\times P\left(\frac{k}{k_3}\right)$

$=\frac{4}{52}\times \frac{3}{51}\times \frac{2}{50}\times \frac{1}{49}$

$=\frac{24}{650740}$

$=\frac{1}{270725}$