某班级学生被询问他们有多少个兄弟姐妹。
结果如表所示。 兄弟姐妹数量的众数是 1。
计算该班级学生人数的最大可能值。 答案 ___ 兄弟姐妹数量的中位数是 2。
计算 x 的最小可能值。 答案 x =___ 兄弟姐妹数量的平均数是 1.525。
计算 x 的值。 答案 x = ___ 飞镖盘由 3 种颜色组成，如下图所示。 一个袋子里装有六个 10 分硬币、三个 20 分硬币和一个 50 分硬币。 Tim 先掷一枚飞镖，然后从袋子里拿出一枚硬币。 画出树状图以展示可能结果的概率。 答案 计算飞镖击中红色区域且拿出的硬币是 20 分的概率。 答案 ___ 写出一个做出的假设。 答案 ___

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1. Use the modal condition to find the greatest possible value of $x$.

The frequency for 1 sibling is $18$.

Since the modal number of siblings is $1$, this frequency must be greater than the frequency for 3 siblings, which is $x$.

So $x<18$.

The greatest possible integer value of $x$ is therefore
$x=17.$

2. Calculate the greatest possible total number of students.

Total number of students
$=5+18+10+17+2$
$=52.$

1. Count how many students have fewer than 2 siblings, then make the median position fall in the group with 2 siblings.

Number of students with $0$ or $1$ sibling
$5+18=23.$
Number of students with $2$ siblings is $10$, so the positions for value $2$ are from the $24$th to the $33$rd student.

For the median to be $2$, the middle position must be at most the $33$rd position.

Total number of students
$=5+18+10+x+2=35+x.$
For the smallest $x$, the total number should first make the median position just exceed the first $23$ values. If $35+x$ is even, the two middle positions must both lie in the $2$ group. Try $x=16$:
$35+16=51.$
The median is the $\frac{51+1}{2}=26$th value, which lies between the $24$th and $33$rd positions.

For $x=15$, total $=50$, and the median is the average of the $25$th and $26$th values, also $2$. Hence the smallest possible value is actually checked by decreasing further until the middle just remains after the 23rd value:

For total $N=46$, the middle two positions are $23$rd and $24$th, giving median $\frac{1+2}{2}=1.5$, not $2$.

Thus the smallest total for median $2$ is $N=47$, where the median is the $24$th value.
$35+x=47$
$x=12.$

1. Write the total number of siblings and total number of students in terms of $x$.

Total number of siblings
$=0(5)+1(18)+2(10)+3(x)+4(2)$
$=46+3x$

Total number of students
$=5+18+10+x+2=35+x$

2. Use the given mean to form and solve an equation for $x$.

Since the mean number of siblings is $1.525$,
$\frac{46+3x}{35+x}=1.525$
$46+3x=1.525(35+x)$
$46+3x=53.375+1.525x$
$1.475x=7.375$
$x=5$

1. Find the probability that the dart hits the red region.

The red sector has angle $120^\circ$ out of $360^\circ$.

$P(\text{Red})=\frac{120}{360}=\frac{1}{3}$

2. Find the probability of picking a 20 cent coin and multiply the independent probabilities.

There are $6+3+1=10$ coins in total, and $3$ are 20 cent coins.

$P(20\text{ cent})=\frac{3}{10}$

Therefore,

$P(\text{Red and 20 cent})=\frac{1}{3}\times\frac{3}{10}=\frac{1}{10}$

1. State a valid probability assumption used in the earlier probability calculation.

A valid assumption is that the dart throw does not affect which coin is picked, so the two events are independent.