Probability
IGCSE Mathematics Topic 8 7:31 English narration · English + 中文 subtitles burned in
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Transcript
Will it rain tomorrow?
明天会下雨吗？
Will the bus be late?
公交车会晚点吗？
Will this die show a six?
这颗骰子会掷出六吗？
Almost nothing in life is certain, and almost nothing is impossible.
生活中几乎没有什么是完全确定的， 也几乎没有什么是完全不可能的。
Most things sit somewhere in between.
大多数事情都介于两者之间。
Mathematics puts a number on maybe, and that number is a probability.
数学给"也许"一个数字，这个数字就是概率。
This is Topic Eight: probability.
这是第八个专题：概率。
You will learn to put a number on chance, to estimate one from an experiment, and to combine two events using three pictures.
你将学会给可能性一个数字，学会用实验来估计概率， 还会用三种图来处理两个事件的组合。
Parts marked Extended are only for the Extended papers.
标注 Extended 的部分只在扩展卷中考查。
Here is the plan.
先看计划。
First the probability scale, and what its numbers mean.
首先是概率标尺，以及上面的数字代表什么。
Then relative and expected frequency, which come out of experiments.
然后是相对频率和期望频数，它们都来自实验。
Then combined events, with one rule for and, and one rule for or.
接着是组合事件：一个"且"的法则，一个"或"的法则。
Last, three pictures.
最后是三种图。
Every probability lives on this one line, from zero to one.
每一个概率都落在这条从零到一的线上。
Zero means impossible: it can never happen.
零表示不可能：它永远不会发生。
One means certain: it must happen.
一表示必然：它一定会发生。
A half sits in the middle, an even chance, like a fair coin.
二分之一在正中间，是均等的机会，就像一枚均匀的硬币。
Below a half we say unlikely; above a half, likely.
小于二分之一我们说不太可能，大于二分之一说很可能。
So an answer of two, or of minus one, is always wrong.
所以答案是二或者负一，一定是算错了。
When every outcome is equally likely, probability is careful counting.
当每一种结果都等可能时，概率就是仔细地数数。
This spinner has eight identical sectors, so each number has the same chance.
这个转盘有八个完全相同的扇形，所以每个数字的机会都一样。
The probability of an event is the number of outcomes you want, divided by the total.
一个事件的概率，等于你想要的结果数除以总数。
So spinning a three is one out of eight, and an even number is four out of eight, which is one half.
所以转到三是八分之一，转到偶数是八分之四，也就是二分之一。
Now a bag with three red counters and five blue ones.
现在有一个袋子，里面有三个红色棋子和五个蓝色棋子。
First, three plus five is eight, so there are eight equally likely outcomes.
先数一数：三加五等于八，所以一共有八种等可能的结果。
So the probability of red is three out of eight.
于是红色的概率是八分之三。
Now, what is the probability of not red?
那么，不是红色的概率是多少？
All the outcomes add up to one, so the chance that something does not happen is one take away the chance that it does.
所有结果加起来等于一，所以某件事不发生的概率，就是一减去它发生的概率。
And check it: there really are five blue out of eight.
再检验一下：蓝色的确实有八个里的五个。
But not every outcome is equally likely.
但并不是每种结果都等可能。
Is this spinner fair?
这个转盘是均匀的吗？
A fair object gives every outcome the same chance; a biased one does not; and random means chance alone decides.
公平的物体让每种结果机会相同；有偏倚的则不然；随机是指结果完全由机会决定。
When you cannot count, you experiment.
当你数不出来时，就去做实验。
The relative frequency is the number of times it happened, divided by the number of trials.
相对频率等于发生的次数除以试验的总次数。
After only ten spins the estimate jumps about wildly.
只转十次时，这个估计值上下乱跳。
After two hundred spins the line settles down close to one value.
转两百次后，曲线就稳定下来，靠近某一个值。
Here are the two frequency ideas side by side.
这里把两个关于频率的概念放在一起。
Relative frequency estimates a probability from an experiment, and it improves the more trials you do.
相对频率用实验来估计概率，试验次数越多越准。
Expected frequency works the other way: you know the probability, and you want to know how often to expect the event.
期望频数则反过来：你已经知道概率，想知道这个事件大约会发生多少次。
Multiply the probability by the number of trials.
用概率乘以试验次数。
So in three hundred rolls you expect fifty sixes.
所以掷三百次，期望出现五十个六。
Real questions ask about two events at once, and there are only two rules.
真实的题目常常同时问两个事件，而法则只有两条。
And, meaning both happen: multiply the probabilities.
"且"，表示两个都发生：把概率相乘。
That works when the events are independent, when the first does not change the second.
这在两个事件独立时成立，也就是第一个不会改变第二个。
Or, meaning either happens: add them.
"或"，表示其中之一发生：把它们相加。
That works when they are mutually exclusive, when they cannot both happen.
这在两个事件互斥时成立，也就是它们不可能同时发生。
Three pictures organise the counting, and here is the first.
有三种图可以帮你整理计数，这是第一种。
A sample space diagram lists every possible outcome in a table.
样本空间图用表格列出所有可能的结果。
Roll two dice and add them: every cell holds a total.
掷两颗骰子并求和：每个格子里是它们的和。
Six times six is thirty-six cells, all equally likely.
六乘六是三十六个格子，每个都等可能。
How many give a total of seven?
有多少个格子的和是七？
Six of them, in a diagonal.
六个，排成一条对角线。
So the probability is six out of thirty-six, which is one sixth.
所以概率是三十六分之六，也就是六分之一。
The second picture is a Venn diagram.
第二种图是维恩图。
The rectangle holds everything that can happen, and each circle is one event.
矩形表示所有可能发生的事，每个圆代表一个事件。
Here, zero point three lies in A alone, zero point one in B alone, and zero point two in the overlap, where both happen.
在这里，零点三只在 A 中，零点一只在 B 中，零点二在重叠部分，表示两者都发生。
Zero point four sits outside.
零点四在圆外。
Add them all and you get one, as you always should.
把它们全加起来等于一，本来就应该这样。
Everything inside either circle is A or B: zero point six.
两个圆里面的全部，是"A 或 B"，加起来是零点六。
The third picture is a tree diagram, and it is the one the exam loves most.
第三种图是树状图，也是考试最爱考的一种。
Each stage becomes a new set of branches.
每一个阶段都变成一组新的分支。
In this bag there are three red and two blue, and a counter is taken twice, put back each time.
这个袋子里有三个红的和两个蓝的， 取两次，每次都放回。
Write a probability beside every branch.
在每条分支旁写上概率。
Then there are only two moves: multiply along a path, and add across paths.
接下来只有两个动作：沿一条路径相乘，把几条路径相加。
Back to our own bag: three red and five blue.
回到我们自己的袋子：三个红的，五个蓝的。
Take one, put it back, take another.
取一个，放回去，再取一个。
Because it goes back, the second draw has the same chances as the first.
因为放了回去，第二次抽取的机会和第一次完全相同。
Multiply along the top path for two reds: three eighths times three eighths is nine sixty-fourths.
沿最上面那条路径相乘，得到两个红的：八分之三乘八分之三等于六十四分之九。
And here is a free check: the four outcomes add up to one.
这里还有一个免费的检验：四个结果的概率加起来等于一。
Change one word and everything changes.
改变一个词，一切都变了。
This time the first counter is not put back.
这一次，第一个棋子不放回去。
Once a red has been taken out, only seven counters are left, and only two of them are red.
拿走一个红色的以后，袋子里只剩七个棋子，其中只有两个是红的。
So the second branch is not three eighths: it is two sevenths.
所以第二条分支不再是八分之三，而是七分之二。
Notice that both the top and the bottom go down.
注意，分子和分母都变小了。
Multiply along the path as before: three eighths times two sevenths is six over fifty-six, which is three over twenty-eight.
像刚才一样沿路径相乘： 八分之三乘七分之二等于五十六分之六，也就是二十八分之三。
One more idea, and it carries the hardest word in the topic: conditional probability.
还有最后一个概念，它带着这个专题里最难的一个词：条件概率。
It means the probability of one event, given that another has already happened.
它指的是在另一件事已经发生的前提下，某个事件发生的概率。
Watch what given really does.
看看"已知"到底做了什么。
Here are a hundred equally likely outcomes, and A happens in thirty.
这里有一百种等可能的结果，其中三十种里 A 会发生。
Now you are told that B happened.
现在告诉你 B 发生了。
Everything outside B disappears, and B holds forty outcomes.
B 之外的一切都消失了，B 里有四十种结果。
Of those forty, A happens in eleven.
在这四十种里，A 发生了十一次。
Let's do one in full.
我们来完整做一道题。
In a class of thirty students, eighteen study French, and seven of those also study German.
一个班有三十名学生，其中十八人学法语， 而这十八人里有七人同时也学德语。
One French student is picked at random. What is the probability that this student also studies German?
现在随机抽到一名学法语的学生， 他也学德语的概率是多少？
Pause here and try it. Ready?
先暂停，自己试一试。
Start with the whole class of thirty.
好了吗？ 先看整个班的三十人。
Eighteen of them study French.
其中十八人学法语。
We are told the student studies French, so the other twelve are out of the picture.
题目告诉我们这名学生学法语，所以另外十二人就不在考虑范围内了。
Look only at those eighteen, and count the ones who also study German.
只看这十八人，数一数其中同时学德语的人。
So the probability is seven out of eighteen, not seven out of thirty.
所以概率是十八分之七，而不是三十分之七。
Four marks students give away.
四个学生常送掉的分。
First, every probability lies between zero and one, and all the outcomes of one experiment add up to one.
第一，每个概率都在零和一之间， 而且一次实验里所有结果的概率加起来等于一。
Second, and means multiply, or means add, and on a tree you multiply along a branch.
第二，"且"用乘，"或"用加，在树状图上沿一条分支相乘。
Third, watch for without replacement: the second fraction changes on the top and on the bottom.
第三，小心"无放回"：第二个分数的分子和分母都会变。
Fourth, expected frequency is the probability times the number of trials.
第四，期望频数等于概率乘以试验次数。
Four quick questions before you go.
走之前来四个小问题。
What does a probability of zero mean?
概率为零表示什么？
Impossible.
不可能。
If the probability of A is zero point three, what is the probability of not A?
如果 A 的概率是零点三，那么"非 A"的概率是多少？
Zero point seven.
零点七。
Do you add or multiply for and?
遇到"且"是加还是乘？
Multiply.
乘。
And along a branch of a tree?
沿着树状图的一条分支呢？
Multiply.
相乘。