During the recent Chinese New Year house spring cleaning, Miss Loi stumbled upon a dusty old diary from years ago …

**2 Jan 199X**

Dear Diary,

This is my first entry.

I woke up at 6.30 AM today. I brushed my teeth at 6.36 AM. After that, I took my bath at 6.39 AM. After that, I ate breakfast at 6.47 AM. After that, I changed into my school uniform at 7.00 AM. After that, I went downstairs to the bus stop to wait for the bus. The bus arrived at 7.05 AM. After that …

*flips pages impatiently and fast-forward several years ahead*

**6 Feb 199(X + 3)**

Dear Diary,

Yeah! I scored full marks again for my math test today! Miss Lim even praised me in front of the class!

At the same time, she also scolded those who failed badly (it’s always the same few guys – one of whom is actually quite cute). She told them they are a disgrace to the school and asked them why can’t they just be like me and study like me.

Actually I hope Miss Lim would stop asking those guys to be like me, because every time she does that, the looks they (and some of my classmates) gave me are pretty scary.

OK I shall stop here – the TCS drama starring the handsome Li Nanxing is about to start soon!

**9 Feb 199(X + 3)**

Dear Diary,

You know Valentine’s Day is around the corner when all my classmates start to

hao lianto me their plans for that night.Anna is already getting all excited about her high class dinner at Jack’s Place. Sue is expecting nothing short of a big bouquet of roses containing the number of stalks symbolizing the number of days since she first met her stead.

As for me, while I would be

VERYhappy if HE (yeah one of the guys who always fails his math tests) asks me out on that day, I don’t think I’d stand a chance especially with this latest pimple outbreak.Well time to study for the Math test which that sadistic Miss Lim had scheduled on 14 Feb.

**10 Feb 199(X + 3)**

Dear Diary,

HE smiled at me at the tuckshop today!!! I looked around just to confirm if HE wasn’t smiling at someone else behind me! I didn’t know how to react but I sure hope he doesn’t think I was acting

dao.Can’t sleep that’s why I’m writing this.

**11 Feb 199(X + 3)**

Dear Diary,

OMG HE smiled at me again today!!! I think I blushed till my pimples burst!

Die

lahdidn’t study much for the math test – all I can think of now are his beautiful eyes and his captivating smile …

**12 Feb 199(X + 3)**

Dear Diary,

You can’t imagine what a life-changing day it’s been! HE APPROACHED ME AT THE TUCKSHOP AND ASKED ME TO BE HIS VALENTINE!!!

Of course I said yes! Can’t wait to tell Anna and Sue about this!

Will go to B B Fashion after school tomorrow to shop for a new dress!

**13 Feb 199(X + 3)**

Dear Diary,

Something terrible has happened.

As HE has been failing

allhis math tests so far, HE told me that Miss Lim has warned him and his parents that tomorrow’s math test is his last chance to stay in our school. If HE fails it he’ll get kicked out of school immediately and our Valentine’s date would be off since HE will most likely be grounded by HIS parents.HE suggested that the only way he could possibly pass the test would be for me to somehow smuggle my answers to him during the test.

I told him that’s

cheating, but I realized it’s either that or no Valentine’s date for us.Should I help him or not? Oh I’m so torn and confused!

**14 Feb 199(X + 3)**

Dear Diary,

It’s Valentine’s Day. It’s also the day of the dreaded math test.

Due to previous incidents of copying, Miss Lim has split the class into separate small rooms of nine students for this test.

As fate would have it, I’m seated directly behind HIM! As I usually finish each test with at least 20 minutes to spare, there should be enough time for me to pass my answers to HIM to copy and then get them back from HIM at the end of the test.

The diagram shows the seating arrangement during the math test. Miss Lim invigilates by patrolling VERY quickly in a general direction from the front to the back of the room.

The only way Miss Lim can spot me passing my answers to HIM (and vice versa) is when she is at the red shaded area in the diagram.

Those rounds when she walks from the back to the front of the room are not considered as it’s too dangerous to pass my answers around when we can’t see her.

Based on my past observations of the paths Miss Lim took in each round of patrol:

- When starting from the front of the room, the probability that she takes the path on her left is
0.6.- At NO point will she travel in the direction back towards the front of the room.
- Whenever she reaches a junction, she NEVER turns toward the direction which immediately ends with a door.
- During each patrol, she NEVER revisits any portion of the path she has already walked
EXCEPTwhen she reaches a dead end, where she turns back along the same path until she reaches the next junction.Whenever she reaches a junction,

- the probability of her going straight is
0.3, and the probability of her turning left is0.4, if she is free from all restrictions described in 2, 3, 4.- the probability of her going straight is
0.6, if she is limited by any of the restrictions imposed in 2, 3, 4.Given that I only have one attempt (right at the start of Miss Lim’s round) to pass HIM my answers, and that HE only has one similar attempt to pass them back to me once HE’s done copying, what’s the probability of either of us getting caught by Miss Lim?

Oh Diary I’m trembling with fear! Please, please let me know my chances of going for my first ever Valentine’s date tonight!

I’ve already bought that dress from B B Fashion. I’ve used up all my pimple cream. I’m DESPERATE!

P.S. Admittedly this question can be a little tedious – hopefully it can help members of the Lonely Hearts Club to pass some time on 14 February 😛

## 7 Comments

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Dear Diary,

Someone on Facebook managed to obtain the 6 paths for Miss Lim to catch the

~~young Miss Loi~~cheating students, and got an answer that's very close to Miss Loi's.To investigate where the answers differ, here's Miss Loi's plot of one of the paths (let's call it Path 1):

Does this coincide with one of yours? Pay attention to the probability at each junction - they depend on the number of choices Miss Lim has at the junction, based on the restrictions described in the question.

P.S. The diagram has been flipped from top to bottom for better visualization from Miss Lim's point of view.

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P(get caught at any 1 attempt) =

(0.6 * 0.6 * 0.4) +

(0.6 * 0.4 * 0.4 * 0.6) +

(0.4 * 0.6 * 0.4 * 0.6) +

(0.4 * 0.4 * 0.4) +

(0.6 * 0.4 * 0.3 * 0.6 * 0.6) +

(0.4 * 0.6 * 0.3 *0.6 * 0.6)

= 0.37504 let the answer be V (for valentine's :P).

After my original answer (guess you know who i am now), I went back to recalculate the answer. I was careless in some junctions, due to the many many conditions which made me a little confused, as well as the fact that i was doing it mentally (i know, no excuse). That was my first mistake.

My second, is one I think allows some thought. This question made me deliberated between multiplying or squaring the original V-value that I got, due to its "special nature". You see, had the students been caught on the first try of cheating, there would not be a second attempt. A student may, on first sight, simply just multiplied V by 2. If so, that student have well fallen into the "trap" of this question. I drew out a probability tree diagram to better "see" the situation. Because I am lazy and I don't know how to attach images, I shall allow Ms Loi to do it for me. But, the gist is, to get the final probability of whether the students will get caught by Ms Lim, one must add V to the to the product of (1-V) (1st attempt successful) and V (2nd attempt failure).

P(getting caught during test)

= V + V(1-V) = 0.609424998 = 0.609 (3sf)

Therefore, I present the answer, 0.609. Or at least what I think is the right answer.

PS: If I'm right, please post the aftermath of the cheating please :3

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BY THE WAY, Ms Loi does not need to cancel off young ms loi at the start of her comments. I would like to quote the start of the post, "During the recent Chinese New Year house spring cleaning, Miss Loi stumbled upon a dusty old diary from years ago …" This gives enough evidence to proof that, it is indeed Ms Loi's love story!!! >:D

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14 Feb 199(X + 3), 5pmI now stand trembling (with HIM by my side) before Miss Lim's desk in the teachers' common room, with Miss Lim's notorious death gaze burning a searing path of guilt through my soul.

Looking back at what happened an hour ago ...

Deriving from the Probabilities lesson taught last week, I understand that the possible paths Miss Loi could take could be represented by a Probability Tree Diagram.

However as we are only interested in those paths that lead to my demise, it's unnecessary to sketch the entire tree but instead recognize that each junction in the classroom corresponds to a node in the tree and the possible directions to take at each junction are like the 'branches' from each node i.e. they are

mutually exclusive eventssince Miss Lim does not possess the 分身术 ability to go straight, left and right at the same time *shudders*Hence using the P(not

E) = 1 − P(E) property described here, we can derive all the probabilities of the directions Miss Lim takes at each junction from the info given in the question, based on the number of possible choices she has at each junction:P.S. It

maynot be obvious to some that whenever there's onlyonepossible choice, P(this choice) = 1duhSo with that settled, I can proceed to plot (with some prompting from a voice from the future) all the possible paths Miss Lim could take to reach my 'danger zone' i.e. Miss Lim's Six Paths to My Damnation.

Since Miss Lim's decision to go straight/turn left/right at each junction is not affected by her decisions at any previous junction, they are all

independent eventsso the probability of each path is computed simply by multiplying her probabilities at all junctions along the way.Now, since I only have one attempt to pass my answers to HIM, I'll get caught in this attempt only if Miss Lim takes Path 1 OR Path 2 OR Path 3 OR Path 4 OR Path 5 OR Path 6 i.e.

P(caught in one attempt)

= 0.144 + 0.0576 + 0.02592 + 0.0576 + 0.064 + 0.02592

= 0.37504

After going through such a long working, some of my mathematically-weaker classmates may be tempted to declare this as the final answer. However there are

twoattempts (i.e. I pass to HIM and HE passes it back to me) that needs to executed which affects the overall probability of being caught.At this junction, I would instantly recognize this as a classic 'two-attempt' probability question that is commonly encountered throughout the TYS with the following probability tree:

As aptly summarised by the freaky voice from the future,

So instead of blindly × 2 or squaring P(caught) like some of my (forever failing) classmates would likely have done, it's important to do some split-second common-sensical analysis and convince yourself it tallies with what's described in the probability tree i.e.

Overall probability of being caught

= P(caught in 1st attempt) OR P(caught in 2nd attempt)

= P(caught in 1st attempt) + P(success in 1st attempt) × P(caught in 2nd attempt)

= 0.37504 + (1 − 0.37504)(0.37504)

≈ 0.609 (3 sig. fig.)

Since there's more than 60% chance of being caught, my mathematical subconsciousness implored me not to do it, but sadly love is blind ... and that's why I'm trembling in front of Miss Lim now 🙁

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14 Feb 199(X + 3), 7pmAfter enduring the most brutal dressing down ever dished out to me in my life, we were finally released from the common room.

Miss Lim will be speaking to our parents soon, but the most hurtful thing to me now is Miss Lim's suspicion that all my previous good results had been achieved through cheating.

Nonetheless I found some small solace in that I'm now walking beside HIM, and took the opportunity to ask him if our Valentine's date tonight is still on, given the chance that Miss Lim might not be calling our parents until tomorrow.

To my utter horror, HE snapped at me

I told him that our chances of success was low, given that the overall probability of getting caught was 0.609 (3 sig. fig.)

With that, he turned his back on me and stormed off.

*cues for camera to zoom away from my face with a rapidly receding background*

It hurt like a thousand roses had just wilted in my heart. I never knew how long I stood on that spot since his departure.

I've failed HIM. And now Miss Lim thinks I've been a cheater my whole life. And I've just used up this month's pocket money on that dress from B B Fashion.

I have never felt so miserable all my life. I …

*CUT!!!*

At this point, Miss Loi slammed shut the pages of the diary following a sudden, chilling realization that HE happens to bear a vague resemblance to someone whom she had met not long ago.曜

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Wait a minute... didn't the young Miss Loi come from an all girls school?

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@mathslover: Dear Diary,

I've flipped through all your pages from front to back (and vice versa) but, for the life of me, couldn't find any mention of me being in an "all girls school".