最近才發現，原來中三數學科，概率 (probability) 課題中提及一個叫「期望值」(expected value) 的概念。
何謂期望值？據中文維基所言，是試驗中每次可能結果的機率 (=概率) 乘以其結果的總和。教科書舉了兩例：
一、估計每250人中，有一人會患上日本腦炎。若市鎮內有750人，問患上日本腦炎人數的期望值是多少？(答案：750 x 1/250 = 3人)
二、口袋裡有一個十元和一個二元硬幣，問隨機抽一個硬幣，期望值是多少？(答案：10 x 0.5 + 2 x 0.5 = 6元)
讀完兩條例題，栗跟學生的反應同是O嘴。首先，患上日本腦炎是壞事，怎麼能「期望」別人患病這麼惡毒？(按：中文只會期望好事) 第二，口袋裡明明只有一個十元和一個二元硬幣，怎麼會抽到個六元出來，變戲法嗎？
問朋友，他說期望值即平均值，唔，算是說得通吧？但栗還是覺得，教科書的例題出得不好，若用買彩券/保險為例，便容易明白得多。另外，是誰想出「期望值」這個劣譯呢？該打。
~ * ~ * ~ * ~ * ~ * ~ * ~ * ~ * ~ * ~
在方老師那邊抄一條數學會考題過來，過過癮：
十次跑步成績，隨機選出兩個，求：
1. 最佳兩次成績當中，只有一次被抽出來
2. 最佳兩次成績當中，最多一次被抽出來
栗的答案：
1. (2/10) x (8/9) x 2 = 16/45
2. 1 - [(2/10) x (1/9)] = 44/45
概率題，一向教栗著迷
expected value 譯作 「 預計值 」 會唔會好啲呢？
回覆刪除[版主回覆04/29/2010 08:28:00]好好多, 不過我發現 '期望值' 已變成公眾認可的譯法, 本本教科書都用...
" 口袋裡明明只有一個十元和一個二元硬幣，怎麼會抽到個六元出來，變戲法嗎？" No, expected values do not necessary correspond to possible outcomes. Just wiki it and everything will become clear in a jiffy. I agree that 期望值 is a poor translation; Isis's translation is much better IMHO.
回覆刪除[版主回覆04/29/2010 08:29:00]不過對於我和學生來說, 用抽銀幣這個例, 好難明呀
Speaking of expected values, check out St Petersburg's Paradox. It will blow your mind.
回覆刪除不過對於我和學生來說, 用抽銀幣這個例, 好難明呀?
回覆刪除Here's an idea: don't look at it as pulling out a coin just once. Tell the kid one is to do it a bunch of time and record the value of the coin each time before throwing it back in. Over a period of N=30+, then take the average of all the values recorded. That's the expected value.
Hey, I just noticed, that problem you got from Mr Fong was in the CEE I took.
回覆刪除[版主回覆04/29/2010 14:14:00]Indeed, Teacher. I got the two test papers yesterday. I'm working out the answers for Paper 2. Hopefully, I'll have it done later today and send it to you ;)
And thanks for your explanation below. Much easier to understand than the example in the textbook!
Are they using this same term 期望值 in China or just in HK? I have no idea what it is without the English term. All these technical terms are so poorly translated. I can't read and understand any Chinese text book. This is what happen if you don't have the ability to lead and be the first on knowledge and technology. You end up following and need to translate other people terminology.
回覆刪除[版主回覆04/29/2010 22:22:00]Not sure about China, but 期望值 is a common translation of 'expected value' in HK. You're definitely right about the translation of technical terms. Most of them are very difficult to understand ...
台灣都係用「期望值」
回覆刪除[版主回覆05/01/2010 06:32:00]你不覺得近年台灣多了很多怪譯嗎？
My explanation of the expected value is not correct. That average would approach the expected value as N approaches infinity. Hmmm, there has to be a simpler way to explain it....
回覆刪除[版主回覆05/01/2010 06:34:00]Teacher: My student understand the concept of infinity. Should be OK for him.
The textbook tried to use a simpler way to explain but failed.
You're arousing my memory on this stuff. I remember once a tutee of mine, way back when I was in U, raised the question of why the expected value of the number of children in US household is 1.2 (a true statistic 40 years ago), but nobody had 1.2 children. He complained that the term "expected value" is a bad choice of word because one doesn't expect that value for that statistic.
回覆刪除
I agree with your friend who said 期望值即平均值 .
By the way, if you enjoy playing around with probabilities and expected values, I strongly recommend you look up the St Petersburg's Paradox. You'll find it fascinating. I guarantee it.
[版主回覆05/02/2010 17:29:00]Teacher: I checked it out as soon as you told me, but the paradox is too complicated for me to understand. I need some time to digest it ... but I haven't got time yet.