一家健身中心有 16 名员工。 随机选择其中 1 名员工。 其为兼职女性员工的概率为 $1/8$。 随机选择其中 2 名员工。 他们均为全职男性员工的概率为 $1/8$。 完成下方关于该健身中心 16 名员工的信息表。 清晰地写出所有辅助计算过程。 答案

1. Find the number of women working part time.

Probability that a randomly selected employee is a woman working part time is $\frac18$.

Since there are 16 employees,
$\text{number of women working part time}=\frac18\times 16=2.$

2. Let the number of men working full time be an unknown and form an equation using probability without replacement.

Let the number of men working full time be $x$.

Two employees are selected at random without replacement. Hence
$\frac{x}{16}\times \frac{x-1}{15}=\frac18.$

3. Solve for the number of men working full time.

$\frac{x(x-1)}{16\times 15}=\frac18$
$x(x-1)=30$
$x^2-x-30=0$
$(x-6)(x+5)=0$

So $x=6$ or $x=-5$.

Since the number of employees cannot be negative, $x=6$.

4. Complete the remaining table entry using the total number of employees.

Known entries:

Women part time $=2$, women full time $=5$, men full time $=6$.

Total accounted for:
$2+5+6=13.$

Therefore men part time:
$16-13=3.$