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# 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.

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• By multiplication law of probability we have $P\big(\large\frac{k}{kkk}\big)$$=P(k)\times P(\large\frac{k}{k})$$\times P(\large\frac{k}{kk})$$\times(\large\frac{k}{kkk}) Let k denote the event that the card drawn is king. We have to find P(k/kkk) Now P(k)=\large\frac{4}{52} P(k/k) is the probability of a second king with the condition that one king has already been drawn.Now there are 3 kings in (52-1)=51cards \therefore P(k/k)=\large\frac{3}{51} P(k/kk) is the probability of a third king with the condition that two kings have already been drawn.Now there are 2 kings in (52-2)=50cards \therefore P(k/kk)=\large\frac{2}{50} P(k/kkk) is the probability that three kings have already been drawn.Now there are 2 kings in (52-3)=49cards \therefore P(k/kkk)=\large\frac{1}{49} Step 2: By multiplication law of probability we have P\big(\large\frac{k}{kkk}\big)$$=P(k)\times P(\large\frac{k}{k})$$\times P(\large\frac{k}{kk})$$\times(\large\frac{k}{kkk})$
$\Rightarrow \large\frac{4}{52}\times\frac{3}{51}\times \frac{2}{50}\times \frac{1}{49}$
$\Rightarrow \large\frac{24}{650740}$
$\Rightarrow \large\frac{1}{270725}$
answered Nov 7, 2013