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Two players play the following game: $A$ writes $3,5,6$ on three different cards: $B$ writes $8,9,10$ on three different cards. Both draw randomly two cards from their collections. Then, $A$ computes the product of two numbers helshe has drawn, and $B$ computes the sum of two numbers he/she has drawn. The player getting the larger number wins. What is the probability that A wins?
$\frac{1}{3}$
$\frac{5}{9}$
$\frac{4}{9}$
$\frac{1}{9}$
Solution
(c)
'Total outcomes of $A$ is $\{(3,5)(3,6)(5,6)\}$
Total outcomes of $B$ is $\{(8,9)(8,10)(9,10)\}$
Case $I$
$A$ wins $A$ get $(3,6)$ and $B$ gets $(8,9)$.
$\therefore$ Probability $=\frac{1}{3} \times \frac{1}{3}=\frac{1}{9}$
Case $II$ $A$ wins $A$ get $(5,6)$ and $B$ gets any outcomes
Probability $=\frac{1}{3} \times 1=\frac{1}{3}$
Total probability $=\frac{1}{9}+\frac{1}{3}=\frac{1+3}{9}=\frac{4}{9}$
