# Quod Erat Demonstrandum

## 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.

P(Chris wins)

$=\frac{1}{10}+\frac{9}{10}\frac{8}{9}\frac{7}{8}\frac{1}{7}$

$+\frac{9}{10}\frac{8}{9}\frac{7}{8}\frac{6}{7}\frac{5}{6}\frac{4}{5}\frac{1}{4}$

$+\frac{9}{10}\frac{8}{9}\frac{7}{8}\frac{6}{7}\frac{5}{6}\frac{4}{5}\frac{3}{4}\frac{2}{3}\frac{1}{2}\frac{1}{1}$

$=\frac{4}{10}=\frac{2}{5}$

C,R,M,C,R,M,C,R,M,C

P(Rachel wins) =$\frac{3}{10}$
P(Mike wins) = $\frac{3}{10}$

$\frac{1}{10}+\frac{1}{10}+\frac{1}{10}+\frac{1}{10}$

P(Chris selects lucky key at the 1st draw);
P(Chris selects lucky key at the 4th draw);
P(Chris selects lucky key at the 7th draw);
P(Chris selects lucky key at the 10th draw)

C,R,M,C,R,M,C,R,M,C

C,R,M,C,R,M,C,R,M,C

{C,R,M,C,R,M,C,R,M,C}

## 4 則迴響 »

1. Suppose that John is a lucky person. Namely, his chance of making a right guess is 2 times higher than an ordinary person, then your student’s argument will not work. It is just an coincidence that your student’s interpretation works.

When I was a secondary school, I drew the tree diagram to solve this type of problem.

迴響 由 Mt — 2012/02/14 @ 11:04 上午 | 回應

2. Can you explain that either distributing the key or picking it on your own are equivalent by the concept of information.
Since the one who receive the key have no any extra information known before they receive all of the keys (i.e. knowing nothing at the beginning). They only know whether they win the car at the moment they try to use the keys.

迴響 由 Justin — 2012/02/14 @ 4:57 下午 | 回應

3. Thank you Mt and Justin!

Justin, it is good to think about information. But students may feel puzzled that when picking up keys one by one, people will try the key immediately, when the lucky guy open the door, no more keys will be picked by the followers. However, when distributing keys, everyone gets a key. So, apparently the situations are different. To persuade them, I just asked them to show the probability that the lucky guy appears at the nth draw is always 1/10.

迴響 由 johnmayhk — 2012/02/14 @ 6:00 下午 | 回應

4. 通告 由 To 5E « Quod Erat Demonstrandum — 2013/02/07 @ 5:09 下午 | 回應