Probability & Statistics 1
A-Level Mathematics Topic 5 11:38 English narration · English + 中文 subtitles burned in
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Roll a single die, and the result is anyone's guess.
掷一颗骰子，结果谁也说不准。
But roll it thousands of times, or measure the heights of thousands of people, and something remarkable happens.
但掷上成千上万次，或者量一量成千上万人的身高， 奇妙的事情就发生了。
The chaos organizes itself.
混乱开始自我组织起来。
A clear, predictable shape emerges from the randomness — often the famous bell curve.
一个清晰、可预测的形状，从随机中浮现出来—— 往往就是那条著名的钟形曲线。
Statistics is the art of finding that hidden order, and probability is how we measure the chance beneath it.
统计学是发现这种隐藏秩序的艺术， 而概率则是我们衡量其背后机会的方式。
This topic has two faces.
这个主题有两副面孔。
First, statistics: taking a flood of data and summarizing it — its centre, its spread, its shape.
第一副是统计：把汹涌的数据加以概括——它的中心、它的离散、它的形状。
Second, probability: counting the ways things can happen, and measuring how likely they are.
第二副是概率：数清事情可能发生的方式，并衡量它们有多大可能。
Together they let us describe the past and predict the future.
两者合在一起， 让我们既能描述过去，又能预测未来。
Let's begin.
让我们开始吧。
Start with raw data.
从原始数据开始。
The first job is to picture it and to summarize it.
第一件事是把它画出来，并加以概括。
A box-and-whisker plot is a favourite: the box holds the middle half of the data, between the lower and upper quartiles, with the median marked inside, and whiskers reaching out to the smallest and largest values.
箱线图是最受欢迎的一种： 箱子装着数据中间的那一半，介于下四分位数和上四分位数之间，中位数标在箱内， 两条须伸向最小值和最大值。
A measure of central tendency is a single middle value: three averages — the mean, the median, and the mode.
要概括中心，你有三种平均——均值、中位数和众数。
A measure of variation shows how spread out the data is: three measures — the range, the interquartile range, and the standard deviation.
要概括离散，你有三种度量——极差、四分位距，以及标准差。
Read a box plot from left to right.
从左到右读箱线图。
The left whisker ends at the lowest value, then comes the lower quartile, then the median line inside the box, then the upper quartile, then the right whisker to the highest value.
左须止于最小值，接着是下四分位数，然后是箱内的中位线， 再是上四分位数，最后右须伸到最大值。
The box itself is the interquartile range — the middle fifty percent of the data.
箱子本身就是四分位距——数据中间百分之五十的部分。
A long whisker or a long box means more spread on that side.
须很长或箱子很长，都说明那一侧更分散。
Two box plots side by side make medians and spreads easy to compare at a glance.
两张箱线图并排，中位数和离散程度一目了然。
A cumulative frequency graph builds running totals.
累积频数图把各频数做成滚动总和。
The curve is usually S-shaped.
曲线通常呈S形。
To estimate the median, find half the total frequency on the vertical axis, draw across to the curve, then drop to the value axis.
要估计中位数，在纵轴上取总频数的一半， 水平画到曲线，再落到数值轴。
Quartiles sit at one quarter and three quarters of the total.
四分位数在总数的四分之一和四分之三处。
A histogram is for grouped data: the area of each bar shows the frequency.
直方图用于分组数据：柱子的面积表示频数。
A stem-and-leaf keeps every original value while showing shape — back-to-back stems compare two sets side by side.
茎叶图保留每一个原始数值并显示形状—— 背靠背的茎可以把两组数据并排比较。
The three averages do not always agree.
三种平均并不总是一致。
Here the data sit at three, five, seven and ten.
这里数据落在三、五、七和十。
The middle value when ordered is five, so the median is five.
排序后中间值是五，所以中位数是五。
Five also appears most often, so it is the mode.
五也出现得最多，所以它是众数。
But the mean is the sum divided by how many there are — thirty divided by five gives six.
但均值是总和除以个数——三十除以五得六。
A high outlier at ten pulls the mean up, while the median and mode stay put.
十这个偏高的离群值把均值往上拉，而中位数和众数不动。
Choose the average that suits the question — and always say which one you used.
选择适合题目的平均—— 并永远说明你用的是哪一种。
Of all the measures of spread, the standard deviation is the most important.
在所有离散度量中，标准差是最重要的。
It measures, on average, how far the values sit from the mean.
它衡量的是，各个值平均而言离均值有多远。
Watch: data huddled close to the mean has a small standard deviation; data flung far and wide has a large one.
看：紧紧挤在均值附近的数据，标准差很小；四处散开的数据，标准差很大。
Square it, and you get the variance.
把它平方，就得到方差。
From just two totals — the sum of the values, and the sum of their squares — you can find both the mean and the standard deviation.
仅凭两个总和——各值之和，以及各值平方之和—— 你就能同时求出均值和标准差。
Worked example.
例题。
Ten values have sum fifty and sum of squares three hundred.
十个值，总和为五十，平方和为三百。
The mean is fifty over ten, which is five.
均值是五十除以十，等于五。
The standard deviation is the square root of three hundred over ten minus five squared — the square root of five, about two point two four.
标准差是三百除以十再减去五的平方，再开方——即根号五，约二点二四。
Coding helps: replace each value by t equals x minus a for an assumed mean a.
编码有帮助：把每个值换成 t 等于 x 减 a，其中 a 是假定平均数。
Then the mean of x is a plus the mean of t, but the standard deviation is unchanged — a pure shift never spreads the data.
于是 x 的均值等于 a 加上 t 的均值，但标准差不变——纯平移不会拉开数据。
To find probabilities, you often need to count arrangements.
要求概率，你常常需要数排列的种数。
Here, order is everything.
在这里，顺序就是一切。
A permutation is an arrangement where order matters — where A then B is different from B then A.
排列是一种讲究次序的安排—— 先甲后乙和先乙后甲是不同的。
A combination is a selection where the order does not matter — where the pair A and B is the same either way.
组合则是一种不讲次序的选取—— 甲和乙这一对，无论怎么排都算同一个。
And when some objects are repeated, like the letters in a word, you divide by the factorial of each repeat count, to avoid over-counting.
而当有些对象重复时，比如一个单词里的字母， 你就要除以每种重复个数的阶乘，以免重复计数。
Look at the picture.
看这张图。
On the left, A B and B A are two different permutations — order matters.
左边，甲乙和乙甲是两种不同的排列——顺序重要。
On the right, the set with A and B is just one combination — swapping the letters does not make a new selection.
右边，甲和乙组成的集合 只是一种组合——交换字母不会产生新的选取。
Exam tip in one sentence: if the question says arrange, line up, or code, think permutations; if it says choose, select, or team, think combinations.
一句话的应试技巧：题目说排列、排队或密码， 就想排列；说选取、选择或队伍，就想组合。
Arranging three out of four objects is four times three times two, which is twenty-four — that is four P three.
从四个对象中排列三个，是四乘三乘二，等于二十四——这就是四选三的排列数。
If order is ignored, divide by three factorial to get four C three, which is four.
若忽略顺序，再除以三的阶乘，得到四选三的组合数，等于四。
So n C r is n P r divided by r factorial.
所以组合数是排列数除以 r 的阶乘。
For NEEDLESS: eight letters, E three times and S twice.
对于 NEEDLESS：八个字母，E 三次、S 两次。
Distinct arrangements equal eight factorial over three factorial times two factorial — three thousand three hundred and sixty.
不同排法等于八的阶乘除以三的阶乘再乘二的阶乘——三千三百六十。
Dice are a clean starting point for probability.
骰子是学概率的干净起点。
Each face is equally likely, so you find a probability by counting: favourable outcomes over total outcomes.
每一面等可能，所以概率就是计数：有利结果数除以总结果数。
The same idea scales up — when outcomes are not listed one by one, count them with permutations or combinations, then form the fraction.
同一思路可以放大——结果无法一一列出时，用排列或组合来数，再写成分数。
Always check that every outcome in the sample space really is equally likely before you count this way.
用这种方法之前，永远先确认样本空间里的每一个结果确实等可能。
Now probability itself.
现在讲概率本身。
A Venn diagram pictures two events as overlapping circles.
维恩图把两个事件画成相互重叠的圆。
For 'A or B', use the addition rule — add the two probabilities, but subtract the overlap, so you don't count it twice.
对于"甲或乙"，用加法法则—— 把两个概率相加，但要减去重叠的部分，以免它被数了两次。
For 'A and B', when the events are independent, use multiplication: simply multiply.
对于"甲且乙"，当两个事件相互独立时， 你只需相乘。
And a conditional probability — the chance of A given that B has already happened — is the overlap divided by the probability of B.
而条件概率——在乙已经发生的前提下甲发生的机会—— 就是重叠部分除以乙的概率。
The shaded lens is the intersection — A and B both happen.
阴影的透镜是交集——甲和乙同时发生。
Events are mutually exclusive if they cannot both happen: the circles do not overlap.
互斥事件不能同时发生：圆不相交。
Independent is different: one event does not change the chance of the other.
独立则不同：一个事件不改变另一个的机会。
Test by checking whether the intersection equals the product of the separate probabilities.
检验方法是看交集是否等于各自概率的乘积。
Worked example: probability of A is zero point five, of B is zero point four, of both is zero point two.
例题：甲的概率零点五，乙的概率零点四，两者同时的概率零点二。
The product is also zero point two, so A and B are independent.
乘积也是零点二，所以甲和乙独立。
A discrete random variable takes separate values, each with its own probability.
离散型随机变量取一些分开的值，每个值都有自己的概率。
List them in a distribution table — and the probabilities must always add up to one.
把它们列进一张分布表—— 而这些概率必须始终加起来等于一。
From the table, two numbers describe it.
从这张表里，两个数就能刻画它。
The expectation, or mean, is each value weighted by its probability, then summed.
期望，也就是均值，是每个值按它的概率加权，再相加。
And the variance measures the spread, in the same style as before.
而方差衡量离散程度， 方式和之前一样。
Two models appear again and again.
有两个模型反复出现。
The binomial distribution counts the number of successes in a fixed number of independent trials — think ten coin tosses.
二项分布数的是：在固定次数的独立试验里成功了多少次—— 想想十次抛硬币。
Its mean is simply the number of trials times the probability of success.
它的均值就是试验次数乘以成功概率。
And the geometric distribution asks a different question: on which trial does the very first success happen?
而几何分布问的是另一个问题： 第一次成功发生在第几次试验？
Its mean is one over the probability of success.
它的均值是一除以成功概率。
Here is the binomial for ten trials with success probability zero point three.
这是十次试验、成功概率零点三的二项分布。
Each bar is the probability of that many successes; the mass clusters around the mean of three.
每根柱子是相应成功次数的概率， 质量聚在均值三附近。
Worked example: probability of exactly two successes.
例题：恰好成功两次的概率。
Take ten choose two, times zero point three squared, times zero point seven to the eight.
取十选二，乘以零点三的平方， 再乘以零点七的八次方。
That is forty-five times zero point zero nine times about zero point zero five seven six five — about zero point two three three.
即四十五乘零点零九乘约零点零五七六五——约零点二三三。
Expectation is ten times zero point three, which is three — on the dashed line.
期望是十乘零点三，等于三——在虚线上。
A Galton board makes the normal distribution visible.
高尔顿板让正态分布看得见。
Balls drop through rows of pins, bouncing left or right at random.
小球穿过一排排钉子，随机向左或向右弹。
At the bottom they pile up — tallest in the middle, thinning toward both ends — into that familiar bell shape.
到底部它们堆积起来——中间最高，两端渐薄——形成那条熟悉的钟形。
Many small random choices, added together, produce a continuous, symmetric distribution.
许多细小的随机选择加在一起，就产生一个连续、对称的分布。
That is why heights, measurement errors, and so many natural totals look normal.
这就是为什么身高、测量误差，以及许多自然总量看起来都呈正态。
The star of continuous data is the normal distribution — that symmetric bell curve.
连续数据的主角是正态分布——那条对称的钟形曲线。
It follows a beautiful rule: about sixty-eight percent of the data lies within one standard deviation of the mean, about ninety-five percent within two, and almost everything within three.
它遵守一条优美的规则： 约百分之六十八的数据落在离均值一个标准差之内，约百分之九十五落在两个标准差之内， 而几乎全部都落在三个标准差之内。
This single pattern describes heights, weights, exam scores, and errors of every kind.
这一个规律，描述了身高、体重、考试分数， 以及各种各样的误差。
To find a probability from the bell curve, you find an area beneath it.
要从钟形曲线求出一个概率，你要找的是它下方的面积。
But every normal curve is different, so we rescale.
但每一条正态曲线都不同， 所以我们重新缩放。
By standardisation — subtracting the mean and dividing by the standard deviation — any normal variable becomes the one standard normal, called Z, whose areas are tabulated.
通过标准化——减去均值，再除以标准差——任何正态变量都变成那唯一的标准正态， 记作标准正态，它的面积已经列成表。
Always sketch and shade first.
永远先画图、先阴影。
And when the numbers are large, the normal even stands in for the binomial, as an approximation, with a small continuity correction.
而当数字很大时， 正态甚至可以代替二项，作为一种近似，再加上一个小小的连续性校正。
A normal probability is an area under the bell.
正态概率是钟形曲线下的面积。
Write X normal with mean mu and variance sigma squared; convert to Z by subtracting the mean and dividing by sigma.
把 X 写成均值为缪、方差为西格玛平方的正态； 减去均值再除以西格玛，换成标准正态。
Worked example: bags of rice have mass X normal with standard deviation zero point one four.
例题：米袋质量为正态，标准差零点一四。
Given probability X less than one point four eight equals zero point two two, find the mean.
已知小于一点四八的概率是零点二二，求均值。
From the table, Z is minus zero point seven seven two.
查表得标准正态值负零点七七二。
So one point four eight minus mu over zero point one four equals that value.
于是一点四八减均值再除以零点一四等于该值。
Rearrange: mu is about one point five nine kilograms, to three significant figures.
整理后均值约一点五九千克，三位有效数字。
When n is large, the binomial bars follow a smooth normal curve with the same mean and variance — n p and n p times one minus p.
当试验次数很大时，二项分布的柱子会贴合一条光滑的正态曲线，均值和方差相同—— 都是 n 乘 p，以及 n 乘 p 乘一减 p。
Because you replace a discrete variable by a continuous one, apply a continuity correction: adjust the boundary by half a unit.
因为你用连续变量代替离散变量， 所以要做连续性校正：边界加减半个单位。
So for at least ten successes, use greater than nine point five on the continuous scale.
于是"至少十次成功"在连续尺度上用大于九点五。
Sketch the bars, draw the matching bell, and correct before you standardise.
先画柱子，再画匹配的钟形，校正之后再标准化。
Before you go, four ways to keep your marks.
结束之前，四个保住分数的办法。
First, decide whether order matters: permutations if it does, combinations if it does not.
第一，先判断顺序是否重要：重要就用排列，不重要就用组合。
Second, for a discrete random variable, always check the probabilities sum to one before using the mean.
第二，对离散型随机变量，用均值之前，一定要检查各概率之和为一。
Third, for the normal distribution, standardise with the z formula, then sketch, shade, and read the table.
第三，做正态分布时，用标准化公式，然后画图、阴影，再查表。
Fourth, remember the continuity correction whenever you approximate a discrete variable by the normal.
第四，凡是用正态去近似一个离散变量，都要记得连续性校正。