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当你猜测时会发生什么

What happens if you guess - Leigh Nataro

概率是一个数学概念,它无处不在。
Probability is an area of mathematics that is everywhere.
我们在天气预报中听到这些
We hear about it in weather forecasts,
比如明天有80%的概率可能会下雪
like there’s an 80% chance of snow tomorrow.
它运用于在体育方面的预测
It’s used in making predictions in sports,
就像确定谁更可能赢得“超级碗”。
such as determining the odds for who will win the Super Bowl.
可能也可以将之运用于帮助设定汽车保险等级
Probability is also used in helping to set auto insurance rates
并且它维持着赌场和彩票行业
and it’s what keeps casinos and lotteries in business.
概率是如何影响你的
How can probability affect you?
让我们来看一个简单的概率问题
Let’s look at a simple probability problem.
能否在一次小测验中随机猜测所有的10个问题
Does it pay to randomly guess on all 10 questions
得到全对或全错
on a true/ false quiz?
换句话说,如果你抛硬币
In other words, if you were to toss a fair coin
10次,然后用它来选择答案,
10 times, and use it to choose the answers,
你得到满分的可能性是多少?
what is the probability you would get a perfect score?
这个看起来很简单。因为每个问题的答案只有两种可能性。
It seems simple enough. There are only two possible outcomes for each question.
但是对于10个问题是对是错
But with a 10-question true/ false quiz,
有很多种可行的方式写下不同的组合
there are lots of possible ways to write down different combinations
Ts和Fs。去理解有多少种不同的结合
of Ts and Fs. To understand how many different combinations,
让我们一起思考更小的正确/错误的考查
let’s think about a much smaller true/ false quiz
你可能会回答,有两个问题。
with only two questions. You could answer
全对,或全错,或者一个对一个错
true true, or false false,or one of each.
先错后对,或者先对后错
First false then true,or first true then false.
因此对于两种可能性的问题有四种不用的方式写出答案
So that’s four different ways to write the answers for a two-question quiz.
那么一次有10种可能性结果的问题呢?
What about a 10-question quiz?
好吧,这次,有太多结果要用手计数并列举。
Well, this time, there are too many to count and list by hand.
为了回答这个问题,我们需要知道基本的运算法则
In order to answer this question, we need to know the fundamental counting principle.
基础运算法则的规定
The fundamental counting principle states
如果对于一个事件有A这种可能的结论
that if there are A possible outcomes for one event,
对于另外的事件有B这种可能性结论,
and B possible outcomes for another event,
所以可以排出B的A次方种结论。
then there are A times B ways to pair the outcomes.
很明显这个法则适用于对或错两种可能结果的问题。
Clearly this works for a two-question true/ false quiz.
对于第一个问题你可能会写下两个不同的答案
There are two different answers you could write for the first question,
并且对于第二个问题你会写下两个不同的答案
and two different answers you could write for the second question.
因而会有2的2次方,或者说是4种方式去写出一次有两种可能性问题的答案。
That makes 2 times 2, or, 4 different ways to write the answers for a two-question quiz.
现在让我们考虑一下一次有十个问题的测验。
Now let’s consider the 10-question quiz.
做这个测验,我们需要延伸基本的运算法则。
To do this, we just need to extend the fundamental counting principle a bit.
我们需要意识到十个问题每一个都可能有两个答案
We need to realize that there are two possible answers for each of the 10 questions.
因此这个结果的可能性是
So the number of possible outcomes is
2×2×2×2×2×2
2, times 2, times 2, times 2, times 2, times 2,
×2×2×2×2
times 2, times 2, times 2, times 2.
或者用简单说法就是2的10次方,
Or, a shorter way to say that is 2 to the 10th power,
这个值是1024。
which is equal to 1,024.
也就是说这是你能写下的所有的Ts和Fs.
That means of all the ways you could write down your Ts and Fs,
只有1024分之1的结论才是完美地符合老师的答案。
only one of the 1,024 ways would match the teacher’s answer key perfectly.
因此你可能通过猜测来得到完美的分数
So the probability of you getting a perfect score by guessing
的比例是1024分之1.
is only 1 out of 1,024,
或者说是千分之一。
or about a 10th of a percent.
很明显,猜并不是一个好办法
Clearly, guessing isn’t a good idea.
事实上,什么会是最普遍的分数呢
In fact, what would be the most common score
如果你和你的朋友一直随意猜测
if you and all your friends were to always randomly guess
10个问题的每个对与错?
at every question on a 10-question true/ false quiz?
好吧,不是每个人都会猜对一半。
Well, not everyone would get exactly 5 out of 10.
但是从长远来看,平均数
But the average score, in the long run,
将会是5。
would be 5.
类似于这种情况,有两个可能的结果
In a situation like this, there are two possible outcomes:
一个问题是正确还是错误,
a question is right or wrong,
这种猜对的可能性
and the probability of being right by guessing
总是1/2。
is always the same: 1/2.
准确得去猜一下平均数
To find the average number you would get right by guessing,
将这种可能性结论相加
you multiply the number of questions
通过猜测得到正确的结论。
by the probability of getting the question right.
因而是10的一半,也就是5.
Here, that is 10 times 1/2, or 5.
研究测试是很有希望地,
Hopefully you study for quizzes,
因为很显然它不要猜测。
since it clearly doesn’t pay to guess.
但是站在这个角度,你可能曾经参加过类似SAT这种标准的测试,
But at one point, you probably took a standardized test like the SAT,
并且大多数人曾经猜过很多问题。
and most people have to guess on a few questions.
这里有20个问题和五个可能的答案
If there are 20 questions and five possible answers
对于每个问题,通过随意猜测得到20个正确答案
for each question, what is the probability you would get all 20 right
的可能性是多少?
by randomly guessing?
那你期望你的分数是多少呢?
And what should you expect your score to be?
让我们用之前的想法。
Let’s use the ideas from before.
首先,因为得到每个正确答案的概率是5分之1,
First, since the probability of getting a question right by guessing is 1/5,
我们希望20个问题中有5分之1的是正确的。
we would expect to get 1/5 of the 20 questions right.
呀,那就是答对四个问题!
Yikes – that’s only four questions!
你认为猜对20个问题的可能性相当小?
Are you thinking that the probability of getting all 20 questions correct is pretty small?
让我们来看一下有多小。
Let’s find out just how small.
你还记得之前基本运算法则的规定么?
Do you recall the fundamental counting principle that was stated before?
一个问题有五个可能的答案,
With five possible outcomes for each question,
我们需要5x5x5x5….
we would multiply 5 times 5 times 5 times 5 times…
好吧,我们可能只是将5作为一个因素
Well, we would just use 5 as a factor
乘以20次,即5的20次方
20 times, and 5 to the 20th power
是95万亿,365十亿,431百万,
is 95 trillion, 365 billion, 431 million,
648千,625.哇,简直巨大!
648 thousand, 625. Wow – that’s huge!
所以随意猜测得到正确答案的概率
So the probability of getting all questions correct by randomly guessing
是95万亿分之1.
is about 1 in 95 trillion.

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