wj:
Now you know that Miss Loi is an EXTREMELY patient tutor.
Miss Loi fell asleep while waiting that night.
While waiting, Miss Loi got abducted by aliens and they released her 12 hours later.

Alright fine! It was a typo!

Thanks for attempting the question to determine how suay my sister was.

To simplify matters, we can first let:

• p be the probability of the parking attendant (P.A.) turning up at any given minute
• q (=1-p) be the probability of the P.A. NOT turning up at any given minute
• X be the number of minutes it takes (starting from 12:05:00PM) for the P.A. to first turn up.

1. For Part 1, as Miss Loi has explained in Comment #12, we're in fact looking for P(X = 4). So you can't simply throw in 1/60 as the answer, since for the P.A. to first appear at 12:08PM, we also have to make sure that she DOESN'T appear at 12:05 AND 12:06 AND 12:07 AND DOES appear at 12:08.

   So your revised workings in Comment #17 are spot-on in reflecting the expression:
   P(X=4) = q × q × q × p = q³p = 0.0158

2. For Part 2, you're right to look for P(appear at 12:05) + … + P(appear at 12:19) as the probability of the P.A. appearing at anytime in the 15-minute range between 12:05-12:19PM.

   However from what we've learned in Part 1, each of these probabilities i.e. P(appear at 12:05), P(appear at 12:06) etc. cannot be 1/60 but are actually:

   P(appear at 12:05) = P(X=1) = q⁰p
   P(appear at 12:06) = P(X=2) = q¹p
   P(appear at 12:07) = P(X=3) = q²p
   .
   .
   .
   P(appear at 12:19) = P(X=15) = q¹⁴p

   Calculating each of these terms is time consuming, but when you sum them all up i.e.

   P(X ≤ 15) = q⁰p + q¹p + q²p + ... + q¹⁴p

   You'll find that P(X ≤ 15) is in effect a Geometric Progression with first term a = p, common ratio r = q and number of terms n = 15.

   Using , P(X ≤ 15) can be simplified to:

   P(X ≤ 15) = = 0.223

   At this point, we should be able to derive the general expression:
   P(X ≤ n) = 1-qⁿ ----- (1)

3. Errr ... Part 3 is actually the opposite of Part 2 - we're looking for the time range in which the probability of the P.A. turning up anytime within this time range is ≤ 10% i.e. P(X ≤ n) ≤ 0.1, where n is the number of minutes from 12:05:00PM.

   So a more appropriate approach would be to make use of expression (1) from Part 2 as follows:

   P(X ≤ n) ≤ 0.1
   1-qⁿ ≤ 0.1
   qⁿ ≥ 0.9
   n lg(59/60) ≥ lg(0.9)
   n ≤ 6.268 ⇒ Max n = 6

   So, the P.A. will have at most a 10% chance of appearing between 12:05:00PM to 12:11PM, which means that Miss Loi still has to tear at 12:15PM (∵ parking coupons are torn at 5-min intervals) Ha ha ha

   P.S. Feel free to use the above expression to obtain other values of n in relation to different probability values. For e.g. you'll find that Miss Loi needs to tear at 12:10PM if she wishes to have at most a 5% chance of being caught - but I don't think she's genetically-programmed to do this.

Just to add further, the above question actually deals with something called a Geometric Distribution (I believe this is currently being taught in some schools but not all) which is used to describe situations that involve a number of independent (success or failure) trials, with the probability p of a successful outcome being the same for each trial.

In all geometric distributions,
P(X ≤ n) = 1 - qⁿ (as shown above)
P(X > n) = qⁿ

1. P(PA first appearing at 12:08) = P(PA not appearing at 12:05-:07) * P(PA appearing at 12:08) = P(PA not appearing at 12:05) * P(PA not appearing at 12:06) * P(PA not appearing at 12:07) * P(PA appearing at 12:08) = 59/60 * 59/60 * 59/60 * 1/60 = 205,739/12,960,000 = 0.0158749 = 1.59%

2. P(appear once between 12:05 and 12:19) = P(appear once at 12:05) + ... + P(appear once at 12:19) = 15 * 1/60 * (59/60)^19 = 0.181658 = 18.17%

3. Since got 18.17% chance hor, then miss loi should tear at 12:15PM lor. Since 1/3 of 18.17% is 6+% 10%.

12:00PM:Target appears - spotted walking towards another car. Miss Loi’s eyes brightens. *Grips steering wheel*"

Wow, you waited more than 12 hours for a parking lot?

cy: 辛灾乐祸!

r0n: Miss Loi has left some comments beneath your workings. Hurry, hurry have a look before Mr Loi comes!

Clarion: Miss Loi passed by a cave just now with many broken boulders scattered around its opening. Were you the one who burst out of it?

2. 1/4 [15min * 1/60 = 15/60 = 1/4]

3. 12:14PM? Coz 6min*1/60 = 6/60 = 1/10 = 10%

But then i think can only tear in 5min intervals. so i guess 12:15PM is more realistic ba. Miss Loi can enlighten anot?