当李老师在周末给他的任何一个学生发短信时，平均每 10 次中大约有 6 次会收到回复。在一个特定的周末，李老师给 25 个学生发了短信。

(i)
结合本题的情境，陈述用二项分布对收到回复的学生人数进行建模所需的两个假设。
(ii)
解释为什么在第 (i) 部分中陈述的其中一个假设在此情境下可能不成立。
(iii)
现在假设这些假设确实成立。清楚地陈述该分布，求至少有一半学生回复的概率。
(iv)
相反，若已知随机选择的一个学生回复的概率为 $p$ 。求最小的 $p$ 值，使得有至少 90% 的概率会有超过 20 个学生回复短信。

二项分布解方程图像法 English

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完整解法

7(i)

1. State the assumption regarding independence of trials in context.

Whether any student replies to the text message is independent of whether any other student replies.

2. State the assumption regarding constant probability of success in context.

The probability of a student replying is constant for each of the 25 students.

答案

Whether any student replies to the text message is independent of whether any other student replies.
The probability of a student replying is constant (at 0.6) for each of the 25 students.

7(ii)

1. Provide a real-world reason why the assumption of independence or constant probability fails in this context.

Explain the failure of independence: Students may communicate with one another about the text message, so one student's decision to reply could affect another's.
Explain the failure of constant probability: Different students have different levels of availability or phone usage habits over the weekend.

答案

Whether a student replies to the text message might not be independent of whether another student replies, as the students might be friends who communicate with each other and decide to reply (or not reply) together.

(Alternatively: The probability of a student replying may not be a constant 0.6, as different students have different phone habits and weekend schedules.)

7(iii)

1. State the probability distribution clearly.

Let $X$ be the number of students who reply out of $25$ .
$X \sim \mathrm{B}(25, 0.6)$

2. Translate 'at least half' into a probability expression and evaluate it.

Half of 25 students is $12.5$ . Since the number of students must be an integer, 'at least half' means 13 or more students.
$\mathrm{P}(X \ge 13) = 1 - \mathrm{P}(X \le 12)$
$\mathrm{P}(X \ge 13) \approx 0.84627 \approx 0.846 \text{ (to 3 s.f.)}$

答案

$0.846$

7(iv)

1. Define the distribution and formulate the required probability inequality.

Let $X$ be the number of students who reply out of 25.
$X \sim \mathrm{B}(25, p)$

We require the probability of more than 20 students replying to be at least $0.90$ :
$\mathrm{P}(X > 20) \ge 0.9$
$\mathrm{P}(X \ge 21) \ge 0.9$
$1 - \mathrm{P}(X \le 20) \ge 0.9$

2. Solve for the critical value of $p$ using a Graphing Calculator (GC).

Using a graphing calculator, we can plot $Y_1 = 1 - \mathrm{P}(X \le 20)$ (where $p$ is the variable $X$ on the GC) and $Y_2 = 0.9$ .
Finding the point of intersection gives:
$p = 0.90377\dots$

3. State the least value of $p$.

Since $\mathrm{P}(X \ge 21)$ is an increasing function of $p$ , we require:
$p \ge 0.90377$

The least value of $p$ is $0.904$ (to 3 s.f.).

答案

$0.904$ (to 3 s.f.)

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