📚 Calculating Probability | 计算概率
Probability is the branch of mathematics that measures the chance of an event happening. We use it every day – when checking a weather forecast, playing a game, or deciding whether to take an umbrella. In mathematics, probability is expressed as a number between 0 and 1, where 0 means an event is impossible and 1 means it is certain. Understanding how to calculate probability helps us make informed predictions and analyse situations involving uncertainty.
概率是数学中衡量某一事件发生可能性的分支。我们每天都会用到它——查看天气预报、玩游戏或决定是否带伞时。在数学中，概率用一个介于 0 和 1 之间的数字表示，0 表示事件不可能发生，1 表示事件必然发生。理解如何计算概率有助于我们做出明智的预测，并分析涉及不确定性的情况。
1. Probability in Everyday Life | 生活中的概率
We constantly encounter probability in our daily routines. When a weather report says there is an 80% chance of rain, that is a probability of 0.8. If you flip a fair coin, the chance of getting heads is 1 in 2, or 0.5. In sports commentary, you might hear that a team has a 'fifty-fifty' chance of winning. Probability allows us to quantify these everyday uncertainties, turning vague notions of likelihood into precise numbers that can be compared and analysed. It also appears in more formal settings, such as insurance companies calculating risk or doctors interpreting test results.
我们在日常生活中经常遇到概率。当天气预报说降雨概率为 80% 时，那就是 0.8 的概率。如果你抛一枚均匀硬币，得到正面的机会是二分之一，即 0.5。在体育评论中，你可能听到某支球队有“五五开”的获胜机会。概率让我们能够量化这些日常的不确定性，将模糊的可能性概念转换为可以比较和分析的精确数字。它也出现在更正式的场合，比如保险公司计算风险或医生解读化验结果。
Being able to calculate probability gives us a powerful tool for decision making. For example, if a bag contains 3 red sweets and 7 blue sweets, the probability of picking a red sweet at random is 3/10, or 0.3. This means that if you pick a sweet many times, you would expect about 30% of your picks to be red. Understanding this helps you gauge how likely a red sweet is before you even put your hand in the bag.
能够计算概率为我们提供了有力的决策工具。例如，如果一个袋子里有 3 颗红色糖果和 7 颗蓝色糖果，随机取出一颗是红色的概率是 3/10，即 0.3。这意味着如果你取很多次，你会期望大约 30% 的取出结果是红色的。理解了这一点，你就能在把手伸进袋子之前判断拿到红色糖果的可能性有多大。
2. The Probability Scale | 概率尺度
The probability scale is a number line from 0 to 1. An event that is impossible, such as rolling a 7 on a standard six-sided die, has a probability of 0. An event that is certain, such as the sun rising tomorrow, has a probability of 1. Most events lie somewhere in between. The closer the probability is to 1, the more likely the event is to happen; the closer to 0, the less likely it is.
概率尺度是一条从 0 到 1 的数轴。不可能发生的事件，例如掷一枚标准六面骰子得到 7 点，其概率为 0。必然发生的事件，例如明天太阳升起，其概率为 1。大多数事件的概率在两者之间。概率越接近 1，事件发生的可能性越大；越接近 0，可能性越小。
We can describe probability using words as well as numbers. A probability of 0 can be described as 'impossible'. A probability very close to 0 might be 'very unlikely', around 0.25 is 'unlikely', 0.5 is an 'even chance', 0.75 is 'likely', and very close to 1 is 'very likely'. The table below summarises these descriptions.
我们既可以用数字也可以用文字来描述概率。概率为 0 可以描述为“不可能”。非常接近 0 的概率可能是“非常不可能”，0.25 左右是“不太可能”，0.5 是“均等机会”，0.75 是“很可能”，非常接近 1 则是“极有可能”。下表总结了这些描述。
| Probability (decimal) | Probability (fraction) | Words |
|---|---|---|
| 0 | 0 | Impossible |
| 0.2 | 1/5 | Very unlikely |
| 0.25 | 1/4 | Unlikely |
| 0.5 | 1/2 | Even chance |
| 0.75 | 3/4 | Likely |
| 0.9 | 9/10 | Very likely |
| 1 | 1 | Certain |
When calculating probabilities, we often express the answer as a fraction, a decimal or a percentage. Percentages are just another way of writing a probability; for example, 0.3 is the same as 30%. At KS3 level, you are expected to be comfortable converting between these three forms when discussing probability.
在计算概率时，我们通常将答案表示为分数、小数或百分数。百分数只是概率的另一种写法；例如，0.3 等同于 30%。在 KS3 阶段，你需要在讨论概率时熟练地在这三种形式之间进行转换。
3. Basic Probability Formula | 基本概率公式
For a situation where all outcomes are equally likely, the probability of an event A happening is given by the formula:
P(A) = Number of favourable outcomes / Total number of possible outcomes
在所有的结果可能性相等的情况下，事件 A 发生的概率由以下公式给出：
P(A) = 有利结果的数量 / 可能结果的总数
if you roll a fair six-sided die, there are 6 possible outcomes: 1, 2, 3, 4, 5, 6. The event "roll an even number" ich simplifies to 1/2, or 0.5. It is crucial to check that all outcomes really are equally likely. A fair coin or fair die has equal probabilities for each outcome, but a bent coin or a biased die does not.
这个公式是计算概率的基础。例如，掷一个均匀的六面骰子，共有 6 种可能的结果：1, 2, 3, 4, 5, 6。事件“掷出偶数”包括三种有利结果：2, 4, 6。因此，P(偶数) = 3/6，化简为 1/2，即 0.5。关键是要检查所有结果是否确实是等可能的。均匀的硬币或骰子每个结果概率相等，但弯曲的硬币或不均匀的骰子则不是。
When using this formula, always write the probability as a fraction in its simplest form, unless the question asks for a decimal or a percentage. Being able to simplify fractions is an essential skill in probability. Sometimes the total number of outcomes is not immediately obvious and you have to count carefully, perhaps by listing all possibilities.
在使用这个公式时，除非题目要求写成小数或百分数，否则始终将概率写成最简分数。会化简分数是概率学习中的一项基本技能。有时结果总数并不明显，你需要仔细数出来，可能需要列出所有可能的情况。
4. Equally Likely Outcomes | 等可能结果
The concept of equally likely outcomes is central to theoretical probability. Equally likely outcomes mean that each individual outcome has the same chance of occurring. Rolling a fair die gives each face a probability of 1/6. Flipping a fair coin gives heads and tails a probability of 1/2 each. If a spinner is divided into equal sectors, each sector is equally likely.
等可能结果的概念是理论概率的核心。等可能结果意味着每个单独的结果发生的可能性相同。掷一个均匀骰子，每个面出现的概率为 1/6。抛一枚均匀硬币，正面和反面的概率各为 1/2。如果一个转盘被分成相等的扇形，每个扇形被转到的可能性相等。
However, many real-life events do not have equally likely outcomes. The probability of it raining tomorrow is not based on the idea that "rain" and "no rain" are equally likely; meteorologists use data and models to estimate the chance. When outcomes are not equally likely, you cannot simply use the basic formula. Instead, you must use experimental data or more advanced rules. In KS3, most problems involve fair dice, coins, well-shuffled cards, and similar fair objects.
然而，许多现实生活中的事件并不具备等可能结果。明天下雨的概率并不是基于“下雨”和“不下雨”等可能的想法；气象学家利用数据和模型来估计机会。当结果不是等可能时，你不能简单地套用基本公式，而必须使用实验数据或更复杂的规则。在 KS3 中，大多数问题涉及均匀的骰子、硬币、洗匀的纸牌及类似的公平对象。
If you are told that a bag contains 5 red and 5 blue counters, then picking a red counter is equally likely to picking a blue counter. If the bag contains 4 red and 6 blue, the outcomes are not equally likely because there are more blue counters. Yet each individual counter is equally likely to be chosen, so you can still use the basic formula by considering the total number of counters.
如果你被告知一个袋子中有 5 个红色和 5 个蓝色筹码，那么拿到红色和蓝色筹码是等可能的。如果袋子里有 4 个红色和 6 个蓝色，那么结果就不是等可能的，因为蓝色筹码更多。但每一个单独的筹码被拿到的可能性是相等的，所以仍然可以通过考虑筹码总数来使用基本公式。
5. Sample Spaces and Listing Outcomes | 样本空间与列举结果
A sample space is the set of all possible outcomes of an experiment. For a single coin flip, the sample space is {Heads, Tails}. For rolling a die, it is {1, 2, 3, 4, 5, 6}. When an experiment combines two or more actions, the sample space can be listed systematically, often using a table or a sample space diagram.
样本空间是一个试验所有可能结果的集合。对于一次硬币抛掷，样本空间是 {正面，反面}。对于掷一个骰子，样本空间是 {1, 2, 3, 4, 5, 6}。当一个试验结合了两个或更多行动时，样本空间可以通过列表的方式系统地呈现，通常使用表格或样本空间图。
ecause the outcomes HT and TH both count as exactly one head. Listing outcomes carefully helps avoid missing any results.
例如，如果你抛两枚硬币，样本空间可以列为 HH、HT、TH、TT，其中 H 表示正面，T 表示反面。注意这里有 4 种等可能的结果。恰好得到一枚正面的概率是 2/4 = 1/2，因为 HT 和 TH 都算作恰好一枚正面。仔细列出结果有助于避免遗漏任何情况。
very useful sample space diagram. The 36 equally likely pairs are shown, and you can easily count favourable outcomes for events such as "the sum is 7" or "both dice show the same number" le to construct and read a sample space table is a key skill in probability at this level.
当掷两个骰子时，一个 6 × 6 的表格是非常有用的样本空间图。它显示了 36 对等可能的结果，你可以轻松地为“和为 7”或“两个骰子显示相同数字”等事件数出有利结果。能够构建和阅读样本空间表是这个阶段概率学习的关键技能。
6. Experimental Probability vs Theoretical Probability | 实验概率与理论概率
er of favourable outcomes / total number of outcomes. Experimental probability, also called relative frequency, is what actually happens when you carry out an experiment or survey. It is calculated as:
Experimental probability = Number of times the event occurs / Total number of trials
理论概率是根据数学，在假设公平条件下应该发生的概率。它使用公式 P(A) = 有利结果数 / 结果总数来计算。实验概率，也称为相对频率，是你实际进行实验或调查时发生的情况。它的计算公式是：
实验概率 = 事件发生的次数 / 试验总次数
l probability often differs slightly from the theoretical value due to randomness, but as the number of trials increases, the experimental probability tends to get closer to the theoretical probability. This is known as the law of large numbers.
例如，如果你掷骰子 100 次，得到六点 18 次，那么掷出六点的实验概率是 18/100 = 0.18。理论概率是 1/6 ≈ 0.167。由于随机性，实验概率常常与理论值略有不同，但随着试验次数的增加，实验概率会趋向于接近理论概率。这被称为大数定律。
Experimental probability is very useful when theoretical probability cannot be easily determined. For example, if you want to know the probability that a football player scores a penalty, you would look at historical data: number of goals scored divided by number of penalties taken. This gives an experimental probability that can be used to estimate future performance. It is important to note that experimental probability is based on actual data, so it changes if more trials are added.
当理论概率不易确定时，实验概率非常有用。例如，如果你想知道一个足球运动员射点球的得分概率，你会查看历史数据：进球数除以主罚点球数。这给出了一个实验概率，可用来估计未来的表现。重要的是要注意，实验概率基于实际数据，如果添加更多试验，它会发生变化。
When comparing experimental and theoretical probabilities, a large number of trials is needed before the relative frequency becomes a reliable estimate of the true probability. This is why scientists repeat experiments many times.
在比较实验概率和理论概率时，需要大量的试验次数才能使相对频率成为真实概率的可靠估计。这就是科学家多次重复实验的原因。
7. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot happen at the same time. For a single roll of a die, the events "roll a 2" and "roll a 5" are mutually exclusive because you cannot roll both a 2 and a 5 on the same roll. However, "roll an even number" and "roll a number greater than 4" are not mutually exclusive, because rolling a 6 satisfies both conditions.
如果两个事件不能同时发生，则它们是互斥的。对于单次掷骰子，“掷出 2”和“掷出 5”这两个事件是互斥的，因为你不可能在同一次投掷中既掷出 2 又掷出 5。然而，“掷出偶数”和“掷出大于 4 的数”并不是互斥的，因为掷出 6 同时满足两个条件。
Understanding whether events are mutually exclusive is important because it affects how we calculate the probability of either event happening. If events are mutually exclusive, the probability of one or the other happening is simply the sum of their individual probabilities. This rule is explored in the next section.
理解事件是否互斥很重要，因为它影响着我们如何计算任一事件发生的概率。如果事件是互斥的，那么其中一个或另一个发生的概率就是它们各自概率的和。这一规则将在下一节探讨。
In everyday language, mutually exclusive events are like two separate paths – you can only take one. For example, when a light turns on, the bulb is either working or faulty; these are mutually exclusive. If you draw one card from a deck, getting a heart and getting a club are mutually exclusive. However, getting a king and getting a heart are not mutually exclusive, because the king of hearts belongs to both.
在日常用语中，互斥事件就像两条分开的路——你只能走一条。例如，当灯亮起时，灯泡要么工作正常要么有故障；这些是互斥的。如果你从一副牌中抽一张，抽到红心和抽到梅花是互斥的。但抽到 K 和抽到红心不是互斥的，因为红心 K 同时属于两者。
8. The Addition Rule for Mutually Exclusive Events | 互斥事件的加法规则
If two events A and B are mutually exclusive, the probability that either A or B occurs is given by:
P(A or B) = P(A) + P(B)
如果两个事件 A 和 B 是互斥的，那么 A 或 B 发生的概率由下式给出：
P(A 或 B) = P(A) + P(B)
y of landing on red is 1/4, and the probability of landing on blue is also 1/4. The events "land on red" and "land on blue" = 1/4 + 1/4 = 1/2. This makes sense because two out of the four colours make up the event.
例如，考虑一个有四个相等扇区的转盘，颜色分别为红、蓝、绿、黄。停在红色的概率是 1/4，停在蓝色的概率也是 1/4。事件“停在红色”和“停在蓝色”是互斥的。因此，P(红色或蓝色) = 1/4 + 1/4 = 1/2。这很合理，因为四种颜色中的两种组成了该事件。
If events are not mutually exclusive, the simple addition rule will double count the outcomes that belong to both A and B. In that case, you must subtract the probability of both occurring. At KS3, you may sometimes meet this with Venn diagram problems, but the focus is mostly on mutually exclusive events where you simply add the probabilities.
如果事件不是互斥的，简单的加法规则会将同时属于 A 和 B 的结果重复计算。在这种情况下，你必须减去两者同时发生的概率。在 KS3 阶段，你有时可能会在韦恩图问题中遇到这种情况，但重点大多在于互斥事件，此时只需将概率相加。
This addition rule is very useful when dealing with probability distributions. For instance, if you know the probabilities of scoring different marks in a test, you can find the probability of scoring either an A or a B by adding the individual probabilities, provided the grades are mutually exclusive (which they are).
这个加法规则在处理概率分布时非常有用。例如，如果你知道在测试中取得不同分数的概率，你可以通过将各自概率相加来求得取得 A 或 B 的概率，前提是这些等级是互斥的（它们确实是）。
9. Complementary Events | 互补事件
The complement of an event A is the event that A does not happen. It is usually written as not A or A'. If the probability of A is P(A), then the probability of not A is:
P(not A) = 1 – P(A)
事件 A 的互补事件是指 A 不发生的事件。通常写作 非 A 或 A'。如果 A 的概率是 P(A)，那么 非 A 的概率是：
P(非 A) = 1 – P(A)
se sometimes it is easier to work out the probability of the complement and subtract from 1. For example, if the probability of winning a game is 0.2, the probability of not winning (losing or drawing, depending on context) is 1 – mplementary events are always mutually exclusive and together cover all possible outcomes.
这个规则非常有用，因为有时先求出互补事件的概率然后再用 1 去减更容易。例如，如果赢得比赛的概率是 0.2，那么不获胜（输或平局，视情况而定）的概率是 1 – 0.2 = 0.8。互补事件总是互斥的，并且一起涵盖所有可能的结果。
When rolling a die, the complement of "rolling a 6" is "not rolling a 6" 3, 4, 5. The probability of rolling a 6 is 1/6, so the probability of not rolling a 6 is 1 – directly counting the favourable outcomes (5 out of 6). Using the complement is especially useful in situations where counting unfavourable outcomes is much easier than counting favourable ones.
掷骰子时，“掷出 6”的互补事件是“没掷出 6”，它包括 1, 2, 3, 4, 5 这些结果。掷出 6 的概率是 1/6，因此没掷出 6 的概率是 1 – 1/6 = 5/6。这通常可以通过直接计数有利结果（6 种中的 5 种）来验证。在计数不利结果远比计数有利结果容易的情况下，使用互补事件特别有帮助。
10. Applying Probability to Solve Problems | 运用概率解决问题
Now we can bring all these ideas together to solve more complex problems. Many probability questions involve a combination of listing outcomes, using the basic formula, and applying the addition rule or complement rule. A systematic approach is essential: identify the sample space, count the total number of outcomes, count the favourable outcomes for the event in question, and then check if any special rules can simplify the calculation.
现在我们可以将这些概念结合起来，解决更复杂的问题。许多概率问题都涉及结合列出结果、使用基本公式，以及应用加法规则或互补规则。系统的方法至关重要：确定样本空间，数出结果总数，数出所问事件的有利结果数，然后检查是否有特殊规则可以简化计算。
nd 5 green marbles. One marble is drawn at random. What is the probability it is not green? green marbles is 5, so P(green) = 5/10 = 1/2. Using the complement rule, P(not green) = 1 – bles: 3 red + 2 blue = 5, so P(not green) = 5/10 = 1/2. Both methods give the same result.
例如，一个袋子里有 3 颗红色、2 颗蓝色和 5 颗绿色弹珠。随机取出一颗。它不是绿色的概率是多少？总数是 10 颗弹珠。绿色弹珠数是 5，所以 P(绿色) = 5/10 = 1/2。使用互补规则，P(非绿色) = 1 – 1/2 = 1/2。或者，数出非绿色的弹珠：3 颗红色 + 2 颗蓝色 = 5，所以 P(非绿色) = 5/10 = 1/2。两种方法结果相同。
sum of the numbers is greater than 8. The sample space has 36 equally likely outcomes. The sums greater than 8 are 9, 10, 11, and 12. List the pairs: sum 9: (3,6), (4,5), (5,4), (6,3) – that's 4 outcomes. Sum 10: (4,6), (5,5), (6,4) – 3 outcomes. Sum 11: (5,6), (6,5) – 2 outcomes. Sum 12: (6,6) – which simplifies to 5/18. Using a sample space table makes this counting straightforward.
另一个经典问题是掷两个骰子。求数字之和大于 8 的概率。样本空间有 36 个等可能结果。大于 8 的和有 9, 10, 11, 12。列出数对：和为 9：(3,6), (4,5), (5,4), (6,3) — 这是 4 个结果。和为 10：(4,6), (5,5), (6,4) — 3 个结果。和为 11：(5,6), (6,5) — 2 个结果。和为 12：(6,6) — 1 个结果。有利结果总数 = 4+3+2+1 = 10。所以概率是 10/36，简化为 5/18。使用样本空间表可使计数变得简单明了。
As a final tip, always check that your probability answers make sense. A probability must never be less than 0 or greater than 1. If you get a fraction like 7/6, you know a mistake has been made. It is also good practice to consider whether your answer can be estimated roughly. If an event seems fairly likely and your calculation gives a probability of 0.1, double-check your counting.
作为最后的提示，务必检查你的概率答案是否合理。概率绝不能小于 0 或大于 1。如果你得到一个像 7/6 这样的分数，你就知道出错了。另一个好习惯是思考你的答案是否可以被大致估算。如果某个事件看起来相当可能发生，而你的计算给出的概率是 0.1，那么要再检查一下你的计数。
Probability becomes more enjoyable when you see it as a tool for solving real puzzles. Whether you are predicting outcomes of games, assessing risk, or simply understanding sports statistics, the ability to calculate probability empowers you to make sense of an uncertain world.
当你把概率视为解决真实谜题的工具时，它会变得更有趣。无论你是在预测游戏结果、评估风险，还是仅仅理解体育统计数据，计算概率的能力都能让你更好地理解这个不确定的世界。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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