Statistical guesswork and incomplete information

Can a simple probability experiment provide a credible basis for more complex decisions?

The key problem: In the absence of essential information, can a simple coin toss be a reasonable approximation for more complex probability experiments like finding the answers to a multiple choice questions? 

Question: Will the number 3 come up when a fair die is thrown? 
Answer: There is a (1/6) probability of the number 3 coming up. How do we know? We arrive at this number for the probability based on a significant amount of background information available to us. This can be summarized as: 

1. The die has 6 faces in total 
2. One of the 6 faces must come up when the die is thrown
3. The number 3 appears on one of these faces and hence, is 1 of the 6 possibilities 
4. Since the die is fair, each face is equally likely to come up 

Alternative scenario 

But what happens if we must answer the question with no background information available to us? Since the question is in “Will…?” form, we can legally respond with a Yes or No and not explain our answer any further. The probability for each answer can be notionally taken as (1/2) in the absence of any further information regarding the possible events. 

Another instance: “Will Spain beat Germany in the football match today?” A football fan (or a researcher with access to historical information) may be able to give a prediction based on how good each team is and how they have played against each other in the past. But for someone who doesn’t follow football at all, a Yes or a No are both equally acceptable answer. The only knowledge he/she needs is that football is a game played between two teams and that one team will win. Neglecting the chance of a draw, for simplicity, both teams have equal probability of winning. This is the same probability as Heads or Tails appearing in a coin toss. It is not uncommon to see people base important decisions on the result of a coin toss, even when the probability of each outcome of those decision is much more complex. 

For example, a Bollywood/Hollywood villain may use a coin toss to decide whether to kill the hero or not. Even though there is a 50% chance of the coin telling him to kill the hero or spare him, the actual decision he needs to take is much more complex. Can we retain the probabilistic simplicity of the coin toss, while applying the principle to more complex statistical problems? Consider the following example. 

Example: A real life situation 

A student is appearing for an examination, where each question offers five choices of answer - A, B, C, D and E. He is unprepared, has no knowledge of the subject and must rely entirely on guesswork to answer the questions. He finds a (fair) coin in his pocket and decides to use it as a tool to get some help from fate. Since he has 5 choices to deal with and the coin has only 2 sides, he devises a simple algorithm. 

1. Is the answer A? Heads: Yes. Stop tossing | Tails: No. Proceed to next step 
2. Is the answer B? Heads: Yes. Stop tossing | Tails: No. Proceed to next step 
3. Is the answer C? Heads: Yes. Stop tossing | Tails: No. Proceed to next step 
4. Is the answer D? Heads: Yes. Stop tossing | Tails: No. Proceed to next step 
5. Is the answer E? If not A, B, C nor D, it must be E 

How likely is he to do well in the examination? Here is a simple analysis. Initially, let us assume that each of the five choices have equal probability to be the right answer to a particular question. This can be reasonably demonstrated by examining the solution of a large number of questions. If 100 questions are taken, 20 each should be A, B, C, D and E. That is the extent of information that is available with the student. 

• Number of choices per answer = 5 
• Number of tosses = 5 – 1 = 4 (according to the algorithm) 

Probability analysis 

Outcome of coin toss				Result	Random Variable (C)	Probability
1	2	3	4
H	-	-	-	A	1	½
T	H	-	-	B	2	¼
T	T	H	-	C	3	1/8
T	T	T	H	D	4	1/16
T	T	T	T	E	5	1/16

• Expected value of C = E(C) = {(1/2) x 1} + {(1/4) x 2} + {(1/8) x 3} + {(1/16) x 4} + {(1/16) x 5} = 1.9375 à 2
• Approximating this as C = 2, this tends to choice B in the sample space
• Variance of C = V(C) = E(C²) – {E(C)}² = S²(C) = 1.4355
• Standard deviation of C = S(C) = 1.1981

Reliability of the coin toss as an estimator 

Intuitively, the above expectation indicates that if the same method is used a large number of times, the student is most likely to get answer B to most questions. Unless a test-creator consciously takes the decision to make each option equally likely, the inherent bias in each human being can be expected to result in an uneven distribution in the solution. 

If a student is very familiar with a professor and his/her past examinations, it is possible to work out a pattern and guess which answer is most likely. Let’s say that a particular professor tends to have 15 As, 20 Bs, 50 Cs, 10 Ds and 5 Es in his solution. In that case, the student may choose to adjust the algorithm to include this incomplete information he has. The simplest way to do this is to re-assign the random variables this: C:1, B:2, A:3, D:4 and E:5. 

Therefore, the probability of getting a C answer using his algorithm is very high.

1 buffet lunch, 8 business lessons

I must have been really distracted. Normally I have a reputation of concentrating the hardest before a table laid with choicest morsels sporting fancy names. Was I getting too caught up in work lately? Or was I always so obsessed with the food as to ignore the little principles that a buffet lunch would teach me? Management, branding, psychology and a little sprinkling of physics – this restaurant had the whole package!

It began simply enough, until we turned left at the final traffic signal. We were there…or so I thought, until we drove right past the huge glass plated building I’d mentally calculated to be right next to the hotel. But where was this place? Huge hoardings announcing their daily specials are plastered all over the city, but where, in God’s name, was the hotel? It wasn’t my first time there, but all the mental landmarks seemed to fail me just then. After driving by yet again, I finally spotted the gate, flanked by newly angled walls restricting the frontage to a good 50% of what I had remembered it to be. I had learnt my first lesson – a practical rendering of the golden rule of managing real estate – Location, location, location!

Having finally made it inside, we managed to stop right in front of a constricted porch that seemed to smirk “Good luck parking!”. We were way too hungry to spend time parking and decided to ‘outsource’ that particular chore, handing over the keys, with a smile, to the valet. And I had learnt my second lesson of the day.

Heading inside, I was startled by my big bro’s piercing ring tone. I glared at him as he picked up his not so compact Smartphone and walked on, still speaking into it. Strolling through the plush hotel lobby I thought I knew so well, with my eyes darting all across the elegant decor, I just didn’t notice his frantic warning gestures. I would’ve walked straight into the elevator door, if the smiling bell boy hadn’t graciously opened it up just in time. I acknowledged him with a nod, while noting down lesson number 3 – always stay one step ahead of your customer.

Before settling down at a corner table, we took a look at the buffet spread. The maître d' politely informed us that soup would be served at the table and took down our preferences. That was my fourth lesson of the day – this time mixing psychology with smart business. The very offer of being served at the table ensures that about 70% of all customers opt for the soup. A very light investment for the hotel, the soup promises to fill a major part of the average stomach. Think of the waiter who gleefully accepts your order to divide two soups by three or one soup by two. There is, of course, little difference between one cup of broth and two, except in the water content. That is just smart business, lesson number 4.

Having fallen for the soup trap, we proceeded to the rest of the buffet. Digging into what looked like a deep tub of Schezwan rice, my spoon soon hit the floor of the pot. Barring seasoned gourmands like yours truly, this little trick often succeeds in defeating the human mind. When presented with a limited quantity of any resource (food, for instance) for self-service, the portion selected by the average human is inversely proportional to the total quantity of the resource perceived to be available. If the quantity of food is on offer is abundant, any man, or woman, would tend to take just what he would need for his/her immediate need. On the other hand, if there is a limited amount to take from, the selfishness factor kicks in and people tend to dig in, hording for a likely future need. In spite of the shallow base, the tub of fried rice gives the illusion of a deep vessel and hence a large quantity of food on offer. Any normal individual would therefore limit his portion to exactly what he/she would need, thus working in favour of the hotel in question. That would be lesson number 5, on intelligent resource planning.

I moved on to the next dish, my mind already struggling to poise these new found sources for knowledge. A closed vessel greeted me, with a polished metallic hemisphere for a lid. An oddly distorted reflection on the convex surface stared back at me. Now, I am no narcissist, but I never knew I was this large! Combine that bit of physics with a reluctant dieter’s psychology and you have just managed to spoil the unfortunate diner’s appetite. I’m not complaining; let’s just call it lesson number 6. And no, it did not keep me away from that barbecued pork chop with pepper sauce.

Digging my fork into the newest addition to my plate, there was one thing I had almost missed. Apart from us, there were exactly four other people in the room: the headwaiter, two waiters and a chef who had stepped in to check out how his creations were selling. Yet, there was absolutely no evident lack of enthusiasm as they hung around, awaiting orders for an extra bowl of ice cream or a bread basket they could bill us for in addition to the buffet lunch. I was reminded of a thoughtful signboard placed in one of the oldest accounts in my own office – “We shall strive to serve you as if you’re still our only client”. I jotted down lesson number 7 which I should have learnt a long time ago: the importance of good customer service, even in the face of adversity.

Big bro and I sat staring eye to eye, with the bill before us, waiting for either of us to break the tension by fishing out some plastic money. He won, but I had a reason. Presenting my IT giant id card, I felt pleased with myself for having got a 20% corporate discount on the total. The little transaction was teaching me another lesson, number 8 – co-branding and customer lock in. On par with the typical conditions of a foodie in the city around noon, I was hungry, willing to pay for food and spoilt for choices. However, this little hotel by the roadside had a strong card to play in its favour – my own corporate ID card! Granted, the bill is considerably bigger than what a ‘regular’ restaurant or club might’ve cost me, but the psychological edge I supposedly gained over the house (read ‘the hotel’) convinces me that this is a clever choice. At the end of the day I get Rs 200/- off, which would’ve in fact got me a decent lunch at most other places. And what does the hotel lose? A mere marketing expense that got them two clients.

As we headed out through the main door, collected the car keys and drove home, I must have left behind a dozen grinning faces unseen at the hotel. But I was having the last laugh. One quick lunch had taught me a bunch of handy lessons, and for once, my eyes were open to spot them all. Now, what’s for dinner?

I, me , myself - The first punch

Come on; look me in the eye if you can. And listen. Yes, destiny loves me, for a reason I am still unsure of. I don’t control my karma, but I have a whole lot of the good variety stacked away somewhere safe. I know what you’re thinking. And I am trying my best to stifle a smile as I type this out.

I’m a quintessential Scorpio, the kind you can recognize at first sight. Love makes me do strange things that work out for the best in astonishing ways. My artwork and my jottings would reflect this. I judge books by their covers and find it to be more accurate than conventional methods. I don’t believe in magic, but a few miracles make my life simply wonderful.

Why would I make promises I can’t keep? Let's just say I'm going with the flow here. I don't know how often I'll update this blog. I don't know how many people will get to read the posts.

But let me give this a try, ok?