Quod Erat Demonstrandum

2015/02/01

黑白球

Filed under: mathematics,NSS — johnmayhk @ 10:33 上午
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同事擬某基本概率題:

In a game, Evan has to draw balls from a bag containing 2 black balls and 3 white balls one by one without replacement. If he gets two consecutive black balls, he wins; otherwise he loses. Find the probability that he wins.

標準答案如下:

P(wins)
=P(BB)+P(WBB)+P(WWBB)+P(WWWBB)
=\frac{2}{5}\frac{1}{4}+\frac{3}{5}\frac{3}{4}\frac{2}{3}+\frac{3}{5}\frac{2}{4}\frac{2}{3}\frac{1}{2}+\frac{3}{5}\frac{2}{4}\frac{1}{3}
=\frac{2}{5}

有沒有留意,盒內共有 5 球,黑球 2 個,\frac{2}{5} 就是從盒取 1 球,得黑球之機會。

試用別的例:盒內共有 7 球,黑球 3 個,Evan 取勝之機會,按標準答案之做法:

\frac{3}{7}\frac{2}{6}+\frac{4}{7}\frac{3}{6}\frac{2}{5}+\frac{4}{7}\frac{3}{6}\frac{3}{5}\frac{2}{4}+\frac{4}{7}\frac{3}{6}\frac{2}{5}\frac{3}{4}\frac{2}{3}+\frac{4}{7}\frac{3}{6}\frac{2}{5}\frac{1}{4}=\frac{3}{7}

看,又是等於在盒子取 1 球,得黑球之機會。

black white balls

這逼使我想:是否存在簡單方法處理原問題及其一般情況?結果如下。 (more…)

2015/01/23

某數算題

Filed under: mathematics,NSS — johnmayhk @ 5:34 下午
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Just reply to a F.5C student on a basic core mathematics question (on P.5.38):

There are 8 outstanding students from junior forms and 9 outstanding students from senior forms in a school this year. 5 out of these 17 students are now selected for an overseas exchange programme. Find the number of combinations of selecting at least 1 student from junior forms and 1 from senior forms.

Here is the ‘so-called’ solution from a student:

_8C_1\times _9C_1\times _{15}C_3

as the student claimed, select 1 from junior, _8C_1 ways; select 1 from senior, _9C_1 ways; then select the remaining 3 students from the remaining 15 students, _{15}C_3 ways, hence, the total number of combination should be _8C_1\times _9C_1\times _{15}C_3, right?

Sorry, it is incorrect. (more…)

2014/02/21

某概率題

Filed under: NSS — johnmayhk @ 3:15 下午
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同事擬一道題:

某作業有 11 題,老師選了 4 題作為家課。

小明沒有記下老師選定的題目,只是隨便找 6 題做之。

問該 6 題包含了老師選定的 4 題之概率。

這是標準題,同事的解如下:

小明在 11 題選 6 題,可有 C_6^{11} 種情況。

該 6 題包含了老師選定的 4 題,另外 2 題可從 11 – 4 = 7 題中選出,共 C_2^7 種情況。

於是,要求的概率為

\frac{C_2^7}{C_6^{11}}

可是,學生給的解如下: (more…)

2014/02/01

某數型

Filed under: Fun — johnmayhk @ 5:59 下午
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擲一顆公平骰子兩次,出現的點數和可以是

2,3,4,…,12

其概率分佈如下:

johnmayhk-loaded-dice

如果該骰子並非公平, (more…)

2014/01/01

HT

Filed under: Additional / Applied Mathematics,Fun,HKALE — johnmayhk @ 12:27 上午
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以下題目也見過數次,現在才知是 1981 年比利時數學競賽的某題目:

不停擲一枚公平硬幣,問以下哪一事件出現的機會較大?

(a) THT 比 TTT 先出現;
(b) TTT 比 THT 先出現。

(T = tail,H = head)

比如

HHTHHHHTHTHHHTT… 就是 (a) 其中一種情況;
HTTHHHTTHHTTTHT… 就是 (b) 其中一種情況;

不知諸君會否認為 (a) 和 (b) 出現之機會均等?

非也, (more…)

2013/02/07

To 5E

Filed under: NSS — johnmayhk @ 5:09 下午
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純粹答學生提問,高手見諒。

Q.1
班中 40 人,其中一人是 John。為玩快活角遊戲,先隨意抽出 5 人。再在 5 人中隨意抽出 1 人做主角。求

(a) John 被抽出玩快活角遊戲之機會;
(b) John 成為快活角遊戲主角之機會。

johnmayhk-hc

(more…)

2012/02/13

Core Math 某題:概率

Filed under: NSS,Teaching — johnmayhk @ 11:14 上午
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In a lucky draw of a car, only 1 key out of 10 can open the door of the car. Chris, Rachel and Mike take turns to draw a key at random without replacement. The person who can open the car door will get the car. Find the probabilities of the following events happening.

(a) Chris will get the car.
(b) Rachel will get the car.
(c) Mike will get the car.

在堂上計算 (a) 時,我寫 (more…)

2010/12/23

to 5D

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 11:32 下午
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Solution to questions (2010-12-22 worksheet)

1.
Toss a fair coin 10 times.

P(exactly 6 “H” and 4 “T”)

= (more…)

2010/06/17

重複數算一例

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 11:05 上午
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擬了一道長題:

第一,我沒有說明該「骰子」的運作:所謂得出的「花」應是指在骰子底部沒有題示出來的那個花!sosad…

第二,明眼人知道我是迆出,因為,用「容斥原理」,兩步KO,變成最多值 4 分的短題,見

http://www.hkms-nss.net/discuz/home/space.php?uid=423&do=blog&id=38

但估計不會有太多同學會用「容斥原理」,所以我 (more…)

2010/06/01

Just a revision on counting

Filed under: Additional / Applied Mathematics,HKALE,Teaching — johnmayhk @ 4:03 下午
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Here is just an ordinary question:

“An elevator starts carrying five persons at the ground floor and then goes up. It can stop at any floor of the building (from the first floor to the third floor). Events that people leaving the elevator are assumed to be independent. Let X be the number of ‘stop’ of the elevator during a “going-up” journey. Find the value of E(X)."

We may use “balls and boxes" (more…)

2010/05/27

just answer a textbook question from my student

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 12:45 下午
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To Lau (my student)

This is the question you’d asked

“A man lives at A and works at C, and is due at work at 8:30 a.m. He always catches the first train from A to B, which is scheduled to arrive at B at 8:15 a.m. Buses leaves B for C every 20 minutes, and the bus which leaves B at 8:20 a.m. is scheduled to arrive outside the factory at C at 8:27 a.m. The train is, on the average, one minute late and has a standard deviation of 4 minutes. The bus always leaves on time, but is, on the average, 1.5 minutes late with a standard deviation of 2 minutes. The man’s employer leaves home in his car at 8:15 a.m. and the time for his journey has mean value 13 minutes with a standard deviation of 3 minutes. Find the probability that the employer arrives before the employee." (more…)

2010/03/27

normal approximation to binomial probabilities

那天上八堂(兩堂中四,兩堂中六,四堂對卷),加午飯及放學開會,再加之前凌晨時份「衝改」模擬試卷(遲下 OSM,即 onscreen marking,老師限時限刻限地點改卷,不能再在凌晨[對於我來說是很好的工作時間]工作了),再再加上有兄弟學校的學生在當天做交換生上我的應數堂,理應準備一些好東西;但一切都因在下極度疲憊忙碌,完全沒有備課;導致當天我想用電腦顯示以 normal distribution 近似 binomial distribution 時,才知學校電腦不能安裝 Java 而不能看到這個模擬。可惜。不過 (more…)

2010/03/14

查表前先查表

Filed under: Additional / Applied Mathematics,Fun,HKALE — johnmayhk @ 6:30 下午
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上兩星期,開始在應用數學堂教常態分佈(normal distribution)。做基礎的「查表」題時,同學第一件事不是在表中查考答案,而是在表中找錯誤!教科書的 normal table 見下: (more…)

2010/01/08

圓桌會意

Filed under: Additional / Applied Mathematics,HKALE,Teaching — johnmayhk @ 4:04 下午
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對中六應數考試卷,討論一道是但噏求其出的題目:

「16 人,當中包括兩對夫婦,隨意被安排坐有 16 座位的圓桌,求該兩對夫婦並排而坐(sit next to each other)之概率。」

設兩對夫婦為 {A_1 , A_2} 及 {B_1 , B_2}。所謂「夫婦並排而坐」,包括(比方說)以下情況

但以下情況卻不是 (more…)

2009/06/26

閒談一些基本東西:導數符號,函數,解釋

1. 高階導數的符號

同學問,為何 D 兩次(即求二階導數)的符號是

\frac{d^2y}{dx^2}

而不是

\frac{dy^2}{dx^2} 或 \frac{d^2y}{d^2x}(more…)

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