Exercise 1.37

Question

A deck of cards is shuffled well. The cards are dealt one by one, until the first time an
ace appears.
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(a) Find the probability that no kings, queens, or jacks appear before the first ace.
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(b) Find the probability that exactly one king, exactly one queen, and exactly one jack
appear (in any order) before the first ace.

fifthist x · Accepted Answer

\begin{enumerate}[label=(\alph*)]
\item  Ignore all the cards except $J,Q,K,A$.  There are $16$ of those, $4$ 
of which are aces.  Each card has an equal chance of being first in the list, 
so the answer is $\frac 14$.

Source: \url{https://math.stackexchange.com/a/3726869/649082}

\item Ignore all the cards except $J,Q,K,A$. There are $4$ choices for a king, 
$4$ choices for a queen and $4$ choices for a jack with $3!$ permutations of the 
cards. Then, there are $4$ choices for an ace. The remaining $12$ cards can be 
permuted in $12!$
ways, so the answer is $\frac{4^{3}	imes3!	imes4	imes12!}{16!}$.

\end{enumerate}

Exercise 1.37

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Exercise 1.37

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