📚 Probability: Calculating Single Event Probability | 概率：单一事件概率的计算
In everyday life, we use words like 'likely', 'unlikely', 'certain' or 'impossible' to describe how probable an event is. In mathematics, probability gives us a more precise way to measure the chance of an event happening, using numbers between 0 and 1. This article covers how to calculate the probability of a single event, interpret probability scales, and avoid common mistakes, following the Cambridge KS3 curriculum.
在日常生活中，我们常用“很可能”、“不太可能”、“必然”或“不可能”来描述事件发生的可能性大小。数学中的概率则用 0 到 1 之间的数字，为我们提供了一种更精确的方式来衡量事件发生的可能性。本文依据剑桥 KS3 课程大纲，介绍如何计算单一事件的概率、理解概率尺度以及避免常见错误。
1. What is Probability? | 什么是概率？
Probability is a measure of how likely an event is to occur. It always takes a value between 0 and 1 inclusive. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain to happen. Values closer to 0 indicate the event is less likely, and values closer to 1 mean it is more likely.
概率是衡量某个事件发生可能性大小的量度。它的取值范围始终在 0 到 1 之间（含 0 和 1）。概率为 0 表示事件不可能发生，概率为 1 表示事件必然发生。数值越接近 0，事件发生的可能性越小；越接近 1，则越可能发生。
2. The Probability Scale | 概率尺度
Probabilities can be placed on a scale from 0 to 1. This scale can also be expressed in fractions, decimals, or percentages. For example, a probability of 0.5 can be written as ½ or 50%, and it represents an event that has an equal chance of happening or not happening, such as getting heads when flipping a fair coin.
概率可以表示在从 0 到 1 的尺度上。这个尺度也可以用分数、小数或百分数表示。例如，概率 0.5 可以写作 ½ 或 50%，它表示一个事件发生与不发生的可能性相等，比如抛一枚均匀硬币得到正面。
The scale is often marked with labels: 'impossible' at 0, 'unlikely' between 0 and 0.5, 'even chance' at 0.5, 'likely' between 0.5 and 1, and 'certain' at 1. Understanding this scale helps us compare different events intuitively.
尺度上通常标记着：0 为“不可能”，0 到 0.5 之间为“不太可能”，0.5 为“等可能”，0.5 到 1 之间为“很可能”，1 为“必然”。理解这个尺度有助于我们直观地比较不同事件的可能性。
3. Basic Probability Formula | 基本概率公式
For a single event where all outcomes are equally likely, the probability can be calculated using a simple formula. You count how many outcomes satisfy the event (favourable outcomes) and divide by the total number of possible outcomes.
对于所有结果等可能发生的单一事件，概率可以用一个简单的公式计算。数出满足事件的结果数（有利结果），然后除以所有可能结果的总数。
P(event) = (Number of favourable outcomes) ÷ (Total number of possible outcomes)
P(事件) = 有利结果数 ÷ 可能结果总数
This formula only works when all outcomes are equally likely, such as rolling a fair dice or spinning a fair spinner. We often simplify the resulting fraction.
只有当所有结果出现的可能性相同时，这个公式才适用，例如掷一枚均匀的骰子或转动一个均匀的转盘。我们通常会化简得到的分数。
4. Calculating Probability for a Fair Die | 公平骰子概率计算
ssible outcomes are 1, 2, 3, 4, 5 and 6. What is the probability of rolling an even number? e are 3 favourable outcomes. The total number of outcomes is 6. Thus, P(even) = 3 ÷ 6 = ½.
设想掷一枚均匀的六面骰子。可能的结果有 1、2、3、4、5 和 6。掷出偶数的概率是多少？偶数为 2、4 和 6，因此有 3 个有利结果。结果总数为 6。因此，P(偶数) = 3 ÷ 6 = ½。
). So P(greater than 4) = 2 ÷ 6 = ⅓. Always remember to simplify fractions where possible.
如果我们想求掷出大于 4 的数的概率，有利结果是 5 和 6（两个结果）。因此 P(大于4) = 2 ÷ 6 = ⅓。请记住，只要可能就应对分数进行化简。
5. Probability of an Event Not Happening | 事件不发生的概率
If the probability of an event happening is P(A), then the probability of the event not happening is 1 − P(A). This is because an event either happens or does not happen, covering all possibilities. This complementary event is often written as A' or 'not A'.
如果事件 A 发生的概率是 P(A)，那么事件不发生的概率为 1 − P(A)。这是因为事件要么发生，要么不发生，涵盖了所有可能性。这个互补事件常写作 A' 或“非 A”。
awing a black card) is 1 − ½ = ½. This rule is extremely useful for checking calculations.
例如，从一副标准的 52 张扑克牌中抽到红色牌的概率是 26/52 = ½。抽不到红色牌（即抽到黑色牌）的概率是 1 − ½ = ½。这条规则对于检查计算结果非常有用。
6. Using the Probability Formula Correctly | 正确使用概率公式
Many students forget to check that all outcomes are equally likely. For instance, in a game with a biased spinner, you cannot simply count sectors unless the spinner is fair. The formula only applies when each outcome has exactly the same chance of occurring.
许多学生忘记了要检查所有结果是否等可能。例如，在一个使用了偏心转盘的游戏中，除非转盘是均匀的，否则你不能简单地按扇区计数。只有每个结果出现的可能性完全相同时，公式才适用。
Another common error is counting outcomes incorrectly. Always list all possible outcomes systematically. For combined events later on, sample space diagrams can help, but for a single event, just be careful to identify every outcome without overlaps.
另一个常见错误是计数结果出错。应该始终系统性地列出所有可能结果。对于后续学到的组合事件，样本空间图能起到帮助作用，但对于单一事件，只要注意无重复地识别每一个结果即可。
7. Probability as Fractions, Decimals and Percentages | 分数、小数和百分数表示的概率
centage. For example, the probability of flipping a coin and getting tails is ½ = 0.5 = 50%. You should be able to convert between these forms fluently.
概率可以用三种等价的形式表示：分数、小数或百分数。例如，抛一枚硬币得到反面的概率是 ½ = 0.5 = 50%。你应该能够熟练地在这些形式之间进行转换。
When a problem asks for an answer in a specific form, make sure you comply. In most KS3 exams, leaving a probability as a simplified fraction is perfectly acceptable, but it is good practice to be comfortable with all three.
当题目要求用特定形式给出答案时，务必遵守。在大多数 KS3 考试中，将概率留作最简分数形式是完全可以接受的，但熟练运用三种形式仍是一个好习惯。
8. Mutually Exclusive Events | 互斥事件
Events are mutually exclusive if they cannot happen at the same time. For example, when rolling a die, the events 'rolling a 2' and 'rolling a 5' are mutually exclusive. For mutually exclusive events, the probability of either A or B occurring is P(A) + P(B).
如果两个事件不能同时发生，则它们是互斥事件。例如，掷骰子时，“掷出 2”和“掷出 5”就是互斥事件。对于互斥事件，事件 A 或 B 发生的概率为 P(A) + P(B)。
This addition rule only works when there is no overlap between the outcomes. Later, you will learn how to handle events that are not mutually exclusive by subtracting the overlap.
这条加法规则仅当结果之间没有重叠时才有效。以后你会学到如何处理并非互斥的事件，即通过减去重叠部分来得出结果。
9. Theoretical vs Experimental Probability | 理论概率与实验概率
The probability we calculate using the formula is called theoretical probability. It tells us what we expect to happen in the long run. Experimental probability is calculated from actual trials or experiments: frequency of the event divided by total number of trials.
我们用公式计算出的概率称为理论概率。它告诉我们在大量重复试验中预期会发生什么。实验概率是通过实际试验或实验计算得出的：事件发生的频数除以试验总次数。
robability is 0.5. As the number of trials increases, experimental probability tends to get closer to theoretical probability.
例如，如果你抛硬币 100 次，得到 53 次正面，那么正面的实验概率为 53/100 = 0.53，尽管理论概率是 0.5。随着试验次数的增加，实验概率往往会越来越接近理论概率。
10. Expected Frequency | 期望频数
robability of rain on a day is 0.3, then over 20 days we expect rain on 0.3 × 20 = 6 days.
我们可以用概率来预测某个事件在一定次数的试验中可能发生的次数。期望频数的计算公式为：期望频数 = 概率 × 试验次数。如果某一天下雨的概率是 0.3，那么在 20 天中，预计下雨的天数为 0.3 × 20 = 6 天。
Remember that expected frequency is only an estimate based on probability; the actual number may differ. This idea helps us check whether observed results are consistent with what we theoretically expect.
请记住，期望频数仅仅是基于概率的估计值，实际发生的次数可能与之不同。这一概念有助于我们检验观察结果是否与理论预期一致。
11. Common Misconceptions and Tips | 常见误区与提示
Do not write probabilities as ratios like '3:6'. Always use a single number between 0 and 1. Also, avoid saying that a probability of ½ means an event will happen once every two trials; it only means that over many trials, it should occur about half the time.
不要将概率写成如“3:6”这样的比式。要始终使用 0 到 1 之间的单个数字。此外，不要认为概率为 ½ 就表示每两次试验事件一定发生一次；它只表明在大量试验中，事件大约有一半的时间会发生。
When using the formula, double-check that you have counted outcomes correctly. Writing a list or drawing a simple table can prevent silly mistakes. Finally, if your calculated probability is greater than 1 or less than 0, you have definitely made an error.
在使用公式时，务必反复检查结果数是否正确。写下清单或绘制简单的表格可以避免粗心错误。最后，如果你计算出的概率大于 1 或小于 0，那一定是出错了。
12. Practice in Context | 置于情境中的练习
ble is taken at random. The total number of marbles is 10. P(red) = 3/10, P(blue) = 2/10 = ⅕, P(green) = 5/10 = ½. Notice that 3/10 + 2/10 + 5/10 = 1, as expected.
设想一个袋子里装有 3 个红色弹珠、2 个蓝色弹珠和 5 个绿色弹珠。随机取出一个弹珠。弹珠总数为 10。P(红色) = 3/10，P(蓝色) = 2/10 = ⅕，P(绿色) = 5/10 = ½。注意 3/10 + 2/10 + 5/10 = 1，符合预期。
What is the probability the marble is not green? P(red) + P(blue) = 3/10 + 2/10 = ½. Working with both approaches reinforces understanding.
弹珠不是绿色的概率是多少？利用互补规则：1 − P(绿色) = 1 − ½ = ½。这与 P(红色) + P(蓝色) = 3/10 + 2/10 = ½ 的和相符。通过两种方法求解能够加深理解。
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