# 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.

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.

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.

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%