두 공 숫자의 곱

숫자 $1$, $2$, $3$이 적혀 있는 공의 개수를 각각 $a$, $b$, $c$라 하면 $a + b + c = 7$ $\displaystyle \mathrm{P} \mathit{\left( X = 4 \right)} = \frac{\mathit{_{b}} \mathrm{C} \mathit{_{2}}}{{} _{7} \mathrm{C} \mathit{_{2}}} = \frac{b \left( b - 1 \right)}{42} = \frac{1}{21}$에서 $b \left( b - 1 \right) = 2$, $b = 2$이므로 $a + c = 5$ $\cdots$$\cdots$ ㉠ $2 \mathrm{P} \mathit{\left( X = 2 \right)} = 3 \mathrm{P} \mathit{\left( X = 6 \right)}$에서 $\displaystyle 2 \times \frac{\mathit{_{a}} \mathrm{C} \mathit{_{1}} \times \mathit{_{b}} \mathrm{C} \mathit{_{1}}}{{} _{7} \mathrm{C} \mathit{_{2}}} = 3 \times \frac{\mathit{_{b}} \mathrm{C} \mathit{_{1}} \times \mathit{_{c}} \mathrm{C} \mathit{_{1}}}{{} _{7} \mathrm{C} \mathit{_{2}}}$, $2 a = 3 c$ $\cdots$$\cdots$ ㉡ ㉠, ㉡에서 $a = 3$, $c = 2$ 따라서 $\mathrm{P} \mathit{\left( X \leq 3 \right)}$$= \mathrm{P} \mathit{\left( X \right.}$$= 1 ) + \mathrm{P} \mathit{\left( X \right.}$$= 2 ) + \mathrm{P} \mathit{\left( X \right.}$$= 3 )$$\displaystyle = \frac{\mathit{_{3}} \mathrm{C} \mathit{_{2}}}{{} _{7} \mathrm{C} \mathit{_{2}}} + \frac{\mathit{_{3}} \mathrm{C} \mathit{_{1}} \times \mathit{_{2}} \mathrm{C} \mathit{_{1}}}{{} _{7} \mathrm{C} \mathit{_{2}}} + \frac{\mathit{_{3}} \mathrm{C} \mathit{_{1}} \times \mathit{_{2}} \mathrm{C} \mathit{_{1}}}{{} _{7} \mathrm{C} \mathit{_{2}}}$$\displaystyle = \frac{5}{7}$