Question

$A$ and $B$ alternately cut a pack of cards which is shuffled after each cut. The game is started by $A$ and continuous until one of the players cuts a club. The probability that $B$ win the game is $k/7$. Find $k$.

Answer 1

Let $A$ be the event that the maximum number on the two chosen tickets is not more that $10$, and $B$ be the event that the minimum number on them is $5$

$\therefore P(A\cap B)=\frac{{}^5{C}_1}{{}^{100}{C}_2}$ and $P\left(A\right)=\frac{{}^{10}{C}_2}{{}^{100}{C}_2}$

Then $P\left(\frac{B}{A}\right)=\frac{P(A\cap B)}{P\left(A\right)}$$=\frac{{}^5{C}_1}{{}^{10}{C}_2}=\frac{1}{9}$