We hear about it in weather forecasts,

我們在天氣預報聽到它，

like there's an 80% chance of snow tomorrow.

像是明天有 80% 的機率會下雪。

It's used in making predictions in sports,

它也被用來作體育預測，

such as determining the odds for who will win the Super Bowl.

像是誰超級盃冠軍的贏率。 （譯註：美國橄欖球賽事。）

Probability is also used in helping to set auto insurance rates

機率還被用來計算汽車保險費，

and it's what keeps casinos and lotteries in business.

讓賭場和樂透繼續經營。

How can probability affect you?

機率對你有什麼影響？

Let's look at a simple probability problem.

看看一個簡單的機率問題。

Does it pay to randomly guess on all 10 questions

如果你隨便猜測十個是非題的答案

on a true/ false quiz?

這樣划算嗎？

In other words, if you were to toss a fair coin

換句話說，如果你正常的投十次硬幣

10 times, and use it to choose the answers,

用它來決定答案，

what is the probability you would get a perfect score?

拿到滿分的機率是多少？

It seems simple enough. There are only two possible outcomes for each question.

每個問題只有兩個可能的解答。

But with a 10-question true/ false quiz,

但是把十個是非題合在一起，

there are lots of possible ways to write down different combinations

就會有一大堆「圈」和「叉」的組合。

of Ts and Fs. To understand how many different combinations,

要了解這麼多種組合，

let's think about a much smaller true/ false quiz

我們從較小的例子想起

with only two questions. You could answer

──只有 2 題是非題。 你可以回答

"true true," or "false false," or one of each.

「圈圈」或是「叉叉」、 或是圈叉各一個。

First "false" then "true," or first "true" then "false."

「先叉後圈」、或是「先圈後叉」。

So that's four different ways to write the answers for a two-question quiz.

共有 4 種方式 來回答這 2 題是非題。

What about a 10-question quiz?

十題是非題呢？

Well, this time, there are too many to count and list by hand.

我們面對數列不完的可能性

In order to answer this question, we need to know the fundamental counting principle.

回答這個問題， 我們必須了解「計數基本原理。」

The fundamental counting principle states

計數基本定理是在說

that if there are A possible outcomes for one event,

如果一個事件有 A 種結果，

and B possible outcomes for another event,

而另一事件有 B 種結果，

then there are A times B ways to pair the outcomes.

那這組事件就會有 A 乘以 B 種結果。

Clearly this works for a two-question true/ false quiz.

很明顯地，兩題是非題上可以這樣做。

There are two different answers you could write for the first question,

第一題你有兩種不同的答案，

and two different answers you could write for the second question.

第二題也有兩種不同的答案。

That makes 2 times 2, or, 4 different ways to write the answers for a two-question quiz.

所以總共就會有 2 以 2， 四種不同的答案。

Now let's consider the 10-question quiz.

現在讓我們重新想想十題是非題。

To do this, we just need to extend the fundamental counting principle a bit.

讓我們延伸一下計數基本原理。

We need to realize that there are two possible answers for each of the 10 questions.

我們知道十題裡的每一題 都有兩種不同的答案。

So the number of possible outcomes is

所以所有可能的答案有

2, times 2, times 2, times 2, times 2, times 2,

2，乘以 2，乘以 2，乘以 2， 乘以 2，乘以 2，

times 2, times 2, times 2, times 2.

乘以 2，乘以 2，乘以 2，乘以 2

Or, a shorter way to say that is 2 to the 10th power,

簡單地說，2 的 10 次方種。

which is equal to 1,024.

也就是 1,024。

That means of all the ways you could write down your Ts and Fs,

這意思是所有你能寫下的圈叉組合

only one of the 1,024 ways would match the teacher's answer key perfectly.

只有其中 1 種 完合符合老師的標準答案。

So the probability of you getting a perfect score by guessing

所以你得到滿分的機率

is only 1 out of 1,024,

只有 1,024 分之 1，

or about a 10th of a percent.

大約是 0.1%。

Clearly, guessing isn't a good idea.

顯然用猜的不是個好主意。

In fact, what would be the most common score

事實上，如果

if you and all your friends were to always randomly guess

在 10 題的是非題裡 你的同學全都用猜的，

at every question on a 10-question true/ false quiz?

那最常出現的分數會是多少呢？

Well, not everyone would get exactly 5 out of 10.

嗯，並不是說每個人都會剛好 在 10 分裡拿到 5 分。

But the average score, in the long run,

但長期看來，平均分數

would be 5.

就會是 5。

In a situation like this, there are two possible outcomes:

在這情況下，有兩種可能

a question is right or wrong,

──某一題是對、或者錯，

and the probability of being right by guessing

而猜對的機率

is always the same: 1/2.

都是一樣 ── 1/2。

To find the average number you would get right by guessing,

如果想要知道平均猜對的題數，

you multiply the number of questions

你可以把題數

by the probability of getting the question right.

乘上一題猜對的機率。

Here, that is 10 times 1/2, or 5.

在這裡，就是 10 乘以 1/2，或是 5。

Hopefully you study for quizzes,

但願你多用點功，

since it clearly doesn't pay to guess.

因為用猜的實在不划算。

But at one point, you probably took a standardized test like the SAT,

不過同時，你可能會想看看 一些像 SAT 的標準考試，

and most people have to guess on a few questions.

大多數人都必須猜個幾題。

If there are 20 questions and five possible answers

如果要猜 20 題、

for each question, what is the probability you would get all 20 right

每題有 5個答案， 那 20 題全猜對的機率

by randomly guessing?

會是多少？

And what should you expect your score to be?

你應該期望你的分數多少？

Let's use the ideas from before.

讓我們用之前的想法想想。

First, since the probability of getting a question right by guessing is 1/5,

首先，因為每題猜對的機率 是 1/5，

we would expect to get 1/5 of the 20 questions right.

我們會期望 20 題裡 有 1/5 的題數會猜對。

Yikes - that's only four questions!

哎呀──只會對 4 題！

Are you thinking that the probability of getting all 20 questions correct is pretty small?

你覺得 20 題要全猜對的機率 非常小嗎？

Let's find out just how small.

我們來看看有多小。

Do you recall the fundamental counting principle that was stated before?

你還記得剛說的 計數基本定理嗎？

With five possible outcomes for each question,

每題有 5 種可能的答案，

we would multiply 5 times 5 times 5 times 5 times...

我們會有 5 乘以 5 成語 5 乘以 5 乘以 ……

Well, we would just use 5 as a factor

呃，我們用 5 當作基數

20 times, and 5 to the 20th power

乘 20 次， 而 5 的 20 次方

is 95 trillion, 365 billion, 431 million,

是 95 兆 3654 億 3164 萬 8625。

648 thousand, 625. Wow - that's huge!

哇，這個數字真大！

So the probability of getting all questions correct by randomly guessing

所以要全部猜對的機率

is about 1 in 95 trillion.

大約是 95 兆分之 1。