Name: 【TED-Ed】猜猜看的結果會如何？ (What happens if you guess - Leigh Nataro)
Uploaded: 2013-10-24T09:35:51.000Z
Duration: 5 min 28 s

Probability is an area of mathematics that is everywhere.

機率是一個隨處可見的數學領域。

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,

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.

Let's look at a simple probability problem.

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

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,

what is the probability you would get a perfect score?

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。

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,

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

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。

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

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

that if there are A possible outcomes for one event,

and B possible outcomes for another event,

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,

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

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。

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。

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，

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,

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

In fact, what would be the most common score

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。

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

and the probability of being right by guessing

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

如果想要知道平均猜對的題數，

by the probability of getting the question right.

在這裡，就是 10 乘以 1/2，或是 5。

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.

If there are 20 questions and five possible answers

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

And what should you expect your score to be?

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。

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？

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,

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 ……

is 95 trillion, 365 billion, 431 million,

So the probability of getting all questions correct by randomly guessing