Monty Hall problem, Gambler’s Fallacy and Male Child

Psychologist Jean Piaget came with concept of “schema”. Schema is a mental framework that helps us to organise and interpret information. These framework help us in dealing with day to day issues ex. we know that if we touch candle flame, we will burn finger, we don’t have to learn this again and again- schema automatically tells us not to touch flame.

While schema is useful (esp. to remain safe in face of danger), it also has drawbacks. It stops you from questioning your existing beliefs and compels you to fall back on time tested old beliefs. It stops new learning, non-linear thinking or coming up with innovative ideas.

Two examples of such rigid thinking are Monty Hall problem and Gambler’s Fallacy.

Monty Hall problem is named after host of TV game show called “ Let us make a deal” ( Indian version is “Khul Ja Sim Sim” with Aman Verma as host). A participant is shown three closed doors, behind one of the doors is grand prize- ex. car, while behind other two doors there is worthless prize ex. goat or stuffed toy fish. A participant is asked to choose one door, the host  makes the game interesting by opening one of the other two doors (it is always door which has worthless prize as host is well aware of what is behind door), now participant has an option either he sticks to his original choice or he swaps. Experiments show that chances of winning car go up significantly if participant swaps.

let us make a deal

Linear or logical mind works like this, in the beginning chances of winning car were 1/3, after one of the door was opened chances of winning are now ½, so sticking to original choice or swap does not make any difference, so one usually sticks to original choice.

Marilyn vos Savant came with different answer, she said that you should always swap. Her explanation was in beginning chances of winning are 1/3, after the one of the doors is opened, the probability of winning for original choice remain 1/3, but probability of winning for other closed door goes up by 2/3. Detailed explanation is available on net. So probabilities of winning are not ½ and ½ they are 1/3 and 2/3. What happens is probability of original choice remains same and has nothing to do with what host does, opening of  second door increases probability of winning for third door.

If instead of 3 there are 1000 doors and there is car behind one door and all other doors have stuffed fish. You chose one door; probability of winning is 1/1000. Now host opens 998 other doors and only one remaining door is unopened, chance of winning prize by choosing this unopened door is 999/1000. Our schema, to simplify things reduce probability to 50-50, but actually probabilities are 1/1000 for original choice and 999/1000 for remaining unopened door.

monty hall

In case of Gambler’s Fallacy, schema tells us that random events will even themselves out over a period of time ex. if gambler is winning 20 times in a row, he will feel he should quit, because he cannot keep winning, after some time he will start losing.

Actually winning or losing in gamble is independent event and has nothing to do with past events ex. he can keep winning next 20 games or lose next 20 games or win next 10 games and lose 10 games thereafter.

gambler's flllacy

There is important application of Gambler’s Fallacy for Indians. Indians prefer son over daughter (if first child is daughter then there is tremendous pressure on mother to give birth to son during next pregnancy). They will try everything to have male child ex. consult astrologer, indulge in rituals, look for favourable day/time to give birth etc.

Gender of child is random event -neither astrologer nor any ritual can predict/change gender of unborn child, so if women wants eight children, all eight will be daughters or all eight can be sons or it can be one son and seven daughters, or 5 sons and 3 daughters or any other combination. So age old blessing of woman giving birth to eight sons – ashta putra saubhagyavati bhava– is meaningless.

Let us go by probability distribution i.e binomial distribution. So probability of giving birth to male child everytime a woman gets pregnant and doing so for 8 times in a row is not even 1%! Most likely scene would be she will have 4 sons and 4 daughters.

No. of sons Probability
0 0.39%
1 3.13%
2 10.94%
3 21.88%
4 27.34%
5 21.88%
6 10.94%
7 3.13%
8 0.39%



Does this salesman really deserve award?

Many organisations have award system to recognise best performers. Some have awards even at functional or departmental level. Most common is awards in sales. It could be star salesman for month, for quarter, for year etc.

Evaluating nominations is time consuming and panel is in sometimes in hurry to declare winner. Deciding winner for Sales is considered to be much simpler as after all, it is all about numbers.

When you declare outstanding salesman of month/quarter/year what do you look at? Who got maximum sales or visually (graphs) that looks like winner or you do statistical analysis?

Panel rarely gets into statistical analysis as telling potential winner that his performance is not outstanding based on statistical analysis is difficult and no manager wants to displease his team member.

Look at example below, salesman A stands out in terms of numbers ( even visually) and should be declared salesman of the quarter. But is the performance really outstanding compared to others or is it just more than others?


For performance to be outstanding it should be statistically significant. One such method is use of ANOVA (Analysis of Variance) for analysing sales performance of all salesmen. The analysis will tell you if performance was really significant. In our case it is not (note calculated F value is less than F critical).

So how many units should salesman A sell to make his performance significant? – Just more unit i.e. 943 units!




David McClelland, IQ and Competencies

“Understanding human motivation ought to be a good thing. It should help us to find out what we really want so that we can avoid chasing rainbows that are not for us.”

-David McClelland, American Psychologist

Are high scores in school/colleges related to high performance in work life?

Most of the management institutes in their entrance exam test students on quantitative ability, reasoning and language ability, which are highly correlated to “g” factor or intelligence factor.

Students with high IQ have higher chances of clearing entrance exam like CAT or XAT. So class of IIM is full of 200+ students with high IQ who compete among themselves. Will this class produce successful leaders? I am not aware of any study correlating grades obtained by IIM student and his success in corporate world, so it is not possible to answer this question.

David McClelland, a psychologist at Harvard, after lot of research came with conclusion that IQ scores and job performance are not related. He also discovered that other traditional measures used in the hiring process, such as examination results and references, were equally poor at predicting success on the job.

McClelland found an alternative to the traditional aptitude and intelligence testing, which yielded a deeper measure that he labelled “competencies.” A competency is defined as an underlying characteristic of a person which enables them to deliver superior performance in a given job, role, or situation i.e. competency separates a superior performer from average performer.


He took sample of superior performers and average performers and using behavioural event interview tried to understand how each of them (i.e. superior and average performer) did their tasks. To avoid bias “double blind” technique was used i.e. interviewer did not know if person was average or superior, while the candidate also did not know if he was interviewed as average or superior.

McClelland went beyond skill and knowledge when he defined competency, competency is a cluster of skill, knowledge, motive, self-image and trait ex. in addition to skill and knowledge a person may have any one of three motives- need to achieve , need for affiliation and need for power. His work and methodology is used by Hays Associates.

Daniel Goleman and Richard Boyatzis came with list of leadership competencies using emotional intelligence frame work.

I will not get into details of competency framework and mapping, as this information is readily available on net.

My point is if McClelland’s research is correct, then entrance exams should be scrapped and selection of students should be done based on behavioural event interview on managerial competencies.





Good Salesman, Bad Salesman and Binomial Distribution

All of you know what good salesman looks like and I am sure you also know how bad salesman looks like. Bad salesman has laundry list of excuses ready for his poor performance…

  • I was not trained properly on products/services
  • Marketing literature, visiting cards, SIM card etc. not given on time.
  • Presales/technical guy did not come with me
  • Did not get enough support from X,Y & Z teams
  • I have been given tough accounts
  • Lack of resources, poor MIS which make me spend lot of time in filling up sales sheet, attendance, outdoor slip etc…

Depending on your organisational culture, such salesmen are either asked to go or put in performance improvement programme with hope that they will improve and manager will not be asked to do unpleasant task of firing poor performer.

During improvement program/counselling session it helps if you tell this person what organisation expectations are and what performance of good salesman looks like.

Binomial distribution is good way of comparing performance of poor, average and good salesman. You can get data of salesmen from your accounts/revenue assurance/ sales coordination department. This will help you to calculate success rate for each category- poor, average and good.


Now suppose they make 10 calls per day and success rates are 30%, 50% and 80% for poor, average and good category. Use excel sheet for calculating binomial distribution, calculations will clearly show difference in performance- for your stars probability of closing 7 or more calls is way higher than that for poor performers (who mostly succeeds in closing 1-4 calls).

Calls Star Solid Pro Deadwood
0 0% 0% 3%
1 0% 1% 12%
2 0% 4% 23%
3 0% 12% 27%
4 1% 21% 20%
5 3% 25% 10%
6 9% 21% 4%
7 20% 12% 1%
8 30% 4% 0%
9 27% 1% 0%
10 11% 0% 0%
Success rate 80% 50% 30%
Mean 8 5 3

If your poor performer has to match performance level of star performer he will have to improve his performance by 266%!

Obedience , “The Lottery” & Organisational Change

“Although the villagers has forgotten the ritual and lost the original black box, they still remembered to use stones.”
– Shirley Jackson, “ The Lottery”

All of us get influenced by society, one such social influence is obedience- act of obeying orders from authority figure. From birth we are used to taking orders from teachers, parents, elders, bosses etc.

Lot of studies have been done on obedience. Starting from Nazism where ordinary Germans, who were loving parents to their children, but actively participated in killing of Jews. Their justification was they were just obeying orders of Nazi leaders.


Some more experiments proved that most of us obey authority against our will and are reluctant to question authority.

In one experiment- Milligram experiment- teachers were asked to administer electric shock to students in case they gave wrong answer, the magnitude of shock varied from 15 volts to 450 volts. Teachers could not see the students but could hear them. While teachers were fully aware than anything above 300 volts was life threatening they did not hesitate to administer shock of very high magnitude- 450 volts to students when they were urged by authorities. Of course the shock was fake, but teachers were not aware of it.


In another experiment- Hofling hospital experiment- 22 nurses were randomly selected and got call from one “Doctor Smith” who asked them give patients 20 mg of fictitious drug called “Astroten”, it was not an approved drug and the label on drug clearly mentioned that maximum dosage was 10 mg. The nurses instead of using logic (ex. checking credentials of doctor, sticking to rule of hospital were only authorised medicine was to be given that too as per prescribed dosage) were ready to give injection to patients, just because orders came from a doctor. The volunteers stopped nurses just before giving injections.


Shirley Jackson wrote a short story called as “The Lottery”. In a village in America, they have a ritual wherein once in a year, they put blank chits in a black box (expect one chit which has black circle on it) and heads of family have to pick up chit from box; family head that gets chit with black circle is eligible for next round. Next round is for members of that family and member who picks up chit with black circle is “winner”- the winner is killed by all the villages with stones! Nobody remembers how that cruel ritual started, what was the reason behind it etc.; they never question it (those who do are censured by village elders), they just blindly followed the ritual. Even when the victim- in this case a lady called Tessie Hutchinson- is getting killed she feels that her selection was unfair, but does not question the ritual itself.


In many organisations status quo is not questioned, because of our tendency to obey authority. That allows authoritative bosses to continue with policies, privileges, processes which no longer benefits organisation (but benefits bosses). Obedience is one of the biggest barriers to organisational change.

Change agents sometimes give up when they start facing resistance from those in senior level who are in favour of status quo. While majority of the employees may agree that they need change, they are unwilling to challenge authority.

ob cartoon

Markov Chains and Your Performance Rating

Russian mathematician Andrey Markov came up with concept of Markov Chains, which has found number of applications in various fields of management. Many professors talk about applying Markov chain to manpower planning/forecasting. I have reservations about using this method, as in today’s dynamic atmosphere it is not of much use. Markov’s transition matrix is useful for companies with very low clock speed and high stability.

But it can be useful in decision making if you understand your company culture and your boss quite well. Depending on your company culture and your boss’s mind-set you can prepare transition matrix which tells your probability of getting different ratings i.e. 50% chance of getting 3, 25% chances of getting 4 and so on.

This is useful for those working in organisations which have poor performance management culture and were performance appraisal is annual ritual delinked from vision of organisation, and your boss decides your ratings before even reading your self- appraisal.


You can start with initial state vector (chances of getting a particular rating when you joined company) and multiply it by transition matrix to know what are your chances of getting different ratings one year hence ex. chances of getting 4 rating next year.

Beauty of Markov Chains is you can use it to predict your ratings for infinite number of years. While the probabilities may change in initial years, in long term probabilities stabilise ex. if your chances of getting 3 ratings in year 1 are 50%, year 2 -55%, Year 3-60% and so on, after 10 years it is say 75% , then it will remain so in 15th year or 20th year or even in 50th year.

After a stage your ratings will stabilise and will not change i.e. you will keep getting same ratings year after year.

If your calculations show that 15 years hence you have 75% chances of getting 3 ratings, 20% chances of getting 2 rating and 5% chances of getting 4 ratings- what decision will you take?