The Philosophy behind the Monty Hall Dilemma

What is the Monty Hall Problem?

The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show  Let's Make a Deal and named after its original host, Monty Hall. The problem was originally posed (and solved) in a letter by Steve Selvin to the American Statistician in 1975.

In the “Monty Hall Dilemma” (MHD), a player tries to guess which of three doors conceals a desirable prize. After an initial choice is made, one of the remaining doors is opened, revealing no prize. The player is then given the option of staying with their initial guess or switching to the other unopened door. Most people opt to stay with their initial guess, despite the fact that switching doubles the probability of winning.

Solution of the Monty Hall Problem using the card method

     Suppose you have 5 cards, like the one shown above. All are heart cards, and you have to choose the Queen of Hearts (QoH). Now, I have turned and shuffled the cards, and you have to pick one. So after you have picked one of the card, which is a random chance and holds 20% probability of having the QoH. I hold the remaining 4 cards. I am the host and I have the power to see my cards, but you can't see your card. There is 80% chance that the QoH is in my hands (for you). 
But for me, I know where the the Queen of Heart is by my knowledge. So if, I throw 3 of my cards, and tell you to choose, between the two cards- the one you have chosen, and the one that is in my hand. It will always be beneficial to you, if you switch to the other card (the one which is in my hand), because there is 80% chance of it being the QoH. It is so that each of the 3 cards shed it 20% chances to the card that I am holding. The probability factor doesn’t get distributed to your card, because no matter what, I can’t remove your card.

This is what is known as relativistic or empirical probability! Knowledge about the past event gives a different probability to the outcome of a factor. It can be observed that the QoH is distributed between the two cards in the ratio of 80:20, to an observer who happened to be all along the game. But for a fresh onlooker the distribution of QoH is 50:50, i.e. either of the two cards can be the QoH.

 It can be simulated to the existence of an electron. Say, half of the QoH exists in my hand and half in the other card which you chose, for a fresh onlooker. But for you 80% of the QoH will be in my hand and 20% in yours (you haven’t seen the card). But for me, 100% of the QoH is in my hand, and none in yours (because I know). So the probability of the existence of the QoH by different observers appears different giving rise to what is known as relativistic probability.

This information brings our own existence and that of our Universe. This is what

Rene Descartes 

meant when he said,

 “Cogito ergo sum – I think, therefore I am.” 

Otherwise we would be a mere manifestation of probability.

This knowledge determines human behaviour, and whole lot of realm of our Universe.

The paradox of distribution of probability

•      The odd thing about the Monty Hall Problem is the way, the probability is distributed after, sharing information.

•      If the hosts knows, which card is the QoH or which door contains the prize, then, after you have chosen the card or door, the host can’t even by random chance remove your choice. So the probability, gets distributed to the other choices (empirical probability).

•      If the host didn’t have any information, then the probability would have been evenly distributed.

•      But then people still stick with their original choice. WHY?

The Greedy Monkey Theory. What is the Story?

How to catch a monkey easily?

Some nuts or banana is kept in a jar with a narrow mouth. So the monkey puts its hand in and grabs as much as it can. Now it tries to withdraw it hands, which gets stuck in the mouth. Now the monkey has a choice, either to lose the nuts or banana, or to lose its freedom. And unfortunately the monkey stays with its initial choice (it will keep the nuts or banana), even though the payoff is bad (for an intelligent observer like us).

That is what happens to most of us, being under the control of emotions and fear. And this is what happens to be close relatives to the monkeys!

Experiments related to Monty Hall Dilemma

·         A series of experiments investigated showed that pigeons (Columba livia) could maximize their expected winnings in a version of the MHD, but not humans. 

·        Replication of the procedure with human participants showed that humans failed to adopt optimal strategies, even with extensive training.

·        Granberg and Brown (1995) gave human participants repeated trials of the MHD, finding that while their participants did become more likely to switch, they did not do so consistently and did not approach the optimal strategy of switching on 100% of trials.

·        When presented repeatedly with two choices, each having a particular likelihood of reinforcement, many species, including humans, tend to match the proportion of their choices to the arranged probabilities.

·        The curious difference between pigeon and human behavior described above might parallel the difference between classical and empirical probability.

·        Pigeons likely use empirical probability to solve the MHD and appear to do so quite successfully. So, Pigeons maximized instead of matching.

·        While humans, tend to solve the problem like fresh onlookers, (the 50:50 approach), and hence tend to perform sub-optimally.

Use of the Monty Hall Problem

•          MHD can be used to observe human behaviour in people investing in Share Market, the economy of strategy. (Whether people will stay or switch to a given share).
•           As discussed earlier, the existence of sub-atomic particles (which brings duality in matter-energy), is a matter of relativistic probability. It exists, because of knowing (knowledge) that it exists. A past event predicting the future, making it happen! And thus conversion from quantum to classical mechanics.
•      Reverse Monty Hall Dilemma can be seen in students solving Multiple Choice Questions (MCQs) in competitive examinations. After ruling out two options (in a set of four options), the students are more likely to make  mistake when they switch their answers between the two options they are unsure about (and most of them do it).
•         The power to think out of the box….. 

Are Birds Smarter Than Mathematicians? Pigeons (Columba livia) Perform Optimally on a Version of the Monty Hall Dilemma