Mathematics is full of fun puzzles. Many puzzles seem so simple yet the solution turns out so counter intuitive. When i first saw this problem i should confess i also fell in the trap of relying on my intuition and i was wrong. I used to wonder why mathematicians are so careful about proofs. Even to prove some theorems whose solution seem obvious, why would the author take such great pain to formulate a long formal mathematical proof? I think, now, i have a little better understanding why so. Our intuition can backfire sometimes in very unimaginable ways but mathematics is always right.
Monty Hall Problem
The problem is basically a probability question and is a puzzle based on an american television game show called "Let's make a deal".
You are in a game show and are presented with three doors. You are told that behind one of these doors is the grand prize (a car) and behind other two, goats. Your goal is to pick the one with the car. So, the host asks you to make a pick. After your pick, the host opens a door with the goat and asks you if you would like to change your choice and pick the other one. What would you do? Would you stay with your previous choice or would you make a switch to the other door?
Some people think it would make no difference at all. I think this mostly comes from intuition. I would have picked the other door that i was about to switch to in the very first place and that would add no more to my knowledge of my chance of winning. From the book "The man who loved only numbers" by Paul Hoffman, " "Physical scientists tend to believe in the idea that probability is attached to things". said Vazsonyi. "Take a coin. You know the probability of a head is 1/2. Physical scientists seem to have the idea that the probability of 1/2 is fused with the coin. Its a property. Its a physical thing. But say i take a coin and toss it a 100 times and each time it comes up tails. You'll say something is wrong, the coin is false but the coin hasn't changed. Its' the same coin that it was when i started to toss it. So, why did i change my mind? because my mind has been upgraded with information. This is the Bayesian view of probability. It took me much effort to understand that the probability is a state of mind " ".
Some people think switching would give an advantage of 1/2 of winning over previous 1/3. I fell into this group when i first saw the question. I was convinced that it would improve my chance but i thought it would improve my chance by 1/2 since there were two doors left out of 3. So, 1 pick out of 2 would give 1/2 probability of winning.
After knowing the real answer to the question, i was puzzled too in the beginning. It was completely off from my intuition. It didn't make sense. How would my chance improve so drastically i.e from 33% to 66%. Only after some serious thought, i could see the logic behind the answer. It turns out the problem of winning is the problem of choosing the wrong door in the very first pick. This is because if you choose the wrong door in the first pick, in the second step you are destined to win because the host will have to pick the one with the goat leaving behind the door with the car. The probability of picking the wrong door in the first pick is 2/3.
This has added further to my conviction to rely more on mathematical approach when intuition collide with reality.
No comments:
Post a Comment