📚 Exercise 18C | 练习18C
Exercise 18C in the IB Mathematics curriculum is designed to strengthen your understanding of discrete random variables, expected value, and variance. This exercise typically appears in the probability distributions chapter and helps you build the skills needed to tackle real IB exam questions. In this article, we will walk through the key concepts, worked examples, common mistakes, and exam tips related to Exercise 18C, ensuring you can solve similar problems with confidence.
练习18C 是 IB 数学课程中旨在强化对离散随机变量、期望值和方差理解的一组题目。该练习通常出现在概率分布章节，帮助您积累应对真实 IB 考题所需的技能。本文将带您梳理与练习18C相关的核心概念、解题示例、常见错误和应试技巧，确保您能自信地解决类似问题。
1. Understanding Discrete Random Variables | 理解离散随机变量
A discrete random variable X takes on a countable number of distinct values, each with an associated probability. For example, the number of heads when tossing three coins can be 0, 1, 2, or 3. The probability distribution lists all possible values of X along with their probabilities, often presented in a table. It is essential that the probabilities sum to 1 and each probability is between 0 and 1.
离散随机变量 X 只能取可数个不同的值，每个值都有一个对应的概率。例如，抛三枚硬币时，正面朝上的次数可以是 0、1、2 或 3。概率分布列出 X 的所有可能取值及其概率，通常以表格形式呈现。必须确保概率之和为 1，且每个概率介于 0 和 1 之间。
nd that the sum of all probabilities equals 1. This foundational step prevents errors later when computing expected value and variance.
在练习 18C 中，您会遇到缺失概率的表格，或需要验证分布是否有效。始终检查 0 ≤ P(X=x) ≤ 1，并且所有概率之和为 1。这个基础步骤可以避免后续计算期望值和方差时出现错误。
2. The Concept of Expected Value | 期望值概念
e context of Exercise 18C, you will often be asked to find E(X) and interpret its meaning.
离散随机变量的期望值 E(X) 表示若将试验重复多次后所得的长期平均结果。其计算方法是将每个取值 x 乘以其概率 P(X=x)，然后求和：E(X) = Σ x · P(X=x)。在练习 18C 中，您常被要求计算 E(X) 并解释其含义。
ich lies between the outcomes. Interpreting this value correctly is an important skill in IB exams.
请注意，期望值不一定是 X 实际能取到的值；它是一个加权平均值。例如，若 X 只能取 1 或 4，各以 0.5 的概率，则 E(X)=2.5，这个数值介于两个可能结果之间。在 IB 考试中，正确解读这一数值是一项重要技能。
3. How to Compute E(X) from a Table | 如何从表格计算E(X)
Consider the following probability distribution for a random variable X:
考虑以下随机变量 X 的概率分布：
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| P(X=x) | 0.2 | 0.3 | 0.4 | 0.1 |
1.2, 4×0.1 = 0.4. Then sum these products: 0.2 + 0.6 + 1.2 + 0.4 = 2.4. Thus, E(X) = 2.4.
计算 E(X) 时，将每个 x 值乘以其对应概率：1×0.2 = 0.2，2×0.3 = 0.6，3×0.4 = 1.2，4×0.1 = 0.4。然后将这些乘积相加：0.2 + 0.6 + 1.2 + 0.4 = 2.4。因此，E(X) = 2.4。
You can set up the calculation in a table with columns for x, P(x), and x·P(x). This structured approach reduces arithmetic mistakes and is highly recommended for Exercise 18C questions.
您可以将计算过程列成表格，栏目依次为 x、P(x) 和 x·P(x)。这种条理化方法能减少算术错误，在处理练习 18C 的题目时强烈推荐。
4. Worked Example: Expected Value in a Game | 实例：游戏中的期望值
with probability 0.3, Y=10 with probability 0.2. Calculate the expected points per round.
在某游戏中，随机变量 Y 表示每轮得分。其概率分布为：Y=0 的概率为 0.5，Y=5 的概率为 0.3，Y=10 的概率为 0.2。计算每轮的期望得分。
1.5 + 2 = 3.5. Therefore, the player can expect to score 3.5 points per round on average.
使用 E(Y) = Σ y P(Y=y) = (0×0.5) + (5×0.3) + (10×0.2) = 0 + 1.5 + 2 = 3.5。因此，玩家平均每轮有望得 3.5 分。
This type of question appears in many IB problem sets because it connects mathematical expectation with real-world decision-making. When you see an Exercise 18C problem about games, remember that expected value helps you determine whether the game is fair or favourable.
这种题型出现在许多 IB 练习中，因为它将数学期望与现实决策联系起来。当您在练习 18C 中遇到游戏类问题时，请记住期望值能帮助您判断游戏是否公平或有利。
5. Introducing Variance and Standard Deviation | 介绍方差与标准差
Variance, denoted Var(X) or σ², measures the spread or dispersion of the random variable's values around the expected value. A low variance indicates values are close to E(X); a high variance indicates greater spread. Standard deviation σ is the square root of variance. In Exercise 18C, you may be asked to compute Var(X) and interpret it.
方差，记作 Var(X) 或 σ²，衡量随机变量取值在期望值周围的分散程度。方差小表示取值集中靠近 E(X)；方差大表示分布较散。标准差 σ 是方差的平方根。在练习 18C 中，您可能需要计算 Var(X
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图