#13 Duck Pond's Probability Problem
Popularity of mental exercises on Chinese social media could be traced to the emphasis on problem solving skills since ancient times.
Picked up a popular post on the Chinese social media today and thought of sharing some insights here. If you like problem solving, I have also provided a write-up on a simple solution based on statistical theory.
Tackling such mental exercises are quite common on the Chinese social media and actually problem-solving skills are highly valued in China since historical times. This is because China has faced a lot of challenges like natural and man-made disasters and also feeding the population has always been a key challenge given that it does not have adequate arable land to feed its humongous population when compared to other countries (see below).
Hence when you look at farming methods in China since ancient times, the Chinese have to employ productive solutions and so problem solving is a routine of the life of many Chinese. One could say that harsh nature has imbibed in the Chinese culture to be pragmatic, and to solve the problems when they arise so as to achieve a good livelihood.
In fact, there is a famous expression from Zen Master Huang Nie 黄櫱 of Tang Dynasty:
“不经一番寒彻骨，怎得梅花扑鼻香”
which means “without the continuous bitter cold, there can be no fragrant plum blossom.”
Thus the Chinese cherish a person’s resilience in the face of difficulties and resolving the problems and not to run away from them would bring the fruits of success eventually.
Problem Description
My colleague’s child is in lower secondary and enrolls in the Mathematical Olympiad class recently. The colleague of mine gave us a problem to do.
(Refer to the diagram).
4 ducks are in a circular pond and each can appear at any point in the pond randomly.
So what is the probability of four ducks appearing together in a half circle?
Our group of four who are masters degree graduates have four different answers, and about to have a fight soon.
Appealing for help at NGA, so what is the answer?
Note : NGA is a community for gamers, so the above is posted at this community.
Solution
Came across a few solutions and some people even resorted to programming (!) as shown below:
Given that the problem is meant for lower secondary student, so one should be able to solve such problems using first principles.
In fact, one needs two probability formulae to solve the problem.
✔️Probability of independent events (A, B, C…) occurring together = (Probability of Event A) multiplied by (Probability of Event B) multiplied (Probability of Event C)…
Example. Probability of getting a head every time during two tosses of the same coin = Probability (first toss is head) = 0.5 x Probability (second toss is head) = 0.5, and so the probability is 0.5x0.5 = 0.25.
✔️ Probability of mutually exclusive events (A, B…) happening = probability of event (A) + probability of event (B) + ….
So to resolve the problem, let us look at duck D1 first and places it at 0° of the diameter XY which splits the circle into two equal halves.
For D1’s event to happen, the fellow ducks would be either above or below XY (all four ducks within the half circle). So the probability of each of the three ducks appearing in either half is ½. So the probability of three ducks in one half that join D1 is (½)x(½)x(½) = 1/[2x3].
Here you can generalise the formula as [1/2 to the power of (n-1)] where n is the total no. of ducks.
But this is only one duck and we have three other ducks for such an event to happen and they are mutually exclusive events.
So the probability for four ducks to be together within a half circle is:
P(D1 event happening) + P (D2 event happening) + …P(D4 event happening) = 4x (½)x(½)x(½) = 4/8 = 0.5.
One could generalise the formula as n x [1/2 to the power of (n-1)].