Probability can be so Unintuitive
Probabilities can
often be totally counter-intuitive. One of the easiest examples to describe is
the Monty Hall problem, named after a game show by the same man:
1)
There
are 3 closed doors. Behind one of them lies a big prize;
2)
You
are asked to pick one. You do so, randomly of course;
3)
At
this point, Monty Hall steps in. He opens one of the doors that you didn’t pick and reveals it to be empty.
He offers you a choice: do you want to switch your choice to the other closed door
or do you want to stick to the one you already picked?
Your intuition
tells you that it doesn’t matter: 2 doors are still closed; so either choice
has a 50-50 chance to win the prize, right?
Wrong! If you
change your choice to the other closed door, you’d have 2/3 chance of winning
whereas you’d have just a 1/3 chance by sticking to your original choice. How
can that be?! And yet it’s true.
Ok, onto our next
counterintuitive concept. If A beats B, and B beats C, then it follows that A
beats C, right?
Not always. If you
arrange the numbers on a dice appropriately, they don’t follow that seemingly
obvious pattern. They’re called “non-transitive
dice”:
In the pic above,
each dice would beat the previous one in the loop on average!
At least the
non-transitive dice is easy to see if you calculate the probabilities. But the
Monty Hall problem is, as Tim
Harford puts it, an “onion of a conundrum; layer after layer, and
guaranteed to make you cry”.
About the logic of "If A beats B, and B beats C, then it follows that A beats C, right?", one can more easily agree with "Not necessarily right". It happens so often is sports, where nearly equal teams compete. Probability theory need not defy common logic in a way here.
ReplyDeleteAbout the previous question, for which the categorical probability declaration has been made thus: "Wrong! If you change your choice to the other closed door, you’d have 2/3 chance of winning whereas you’d have just a 1/3 chance by sticking to your original choice", doesn't seem to hold water. I just can't imagine why simply changing from the made-choice should change the probability.
It's like this: a person makes a choice initially. The probability of 1/3 is fine, but there is this reality that the choice happened to be the Prize, just by fluke. Probability has already turned into the reality. That's certainly possible. Now, someone opens one of the not-selected boxes. Sure enough the prize is absent in that box. How can this truth be defied by any theory - probability or otherwise - "the prize is inside the selected box which is a reality. How can the prize shift to the other remaining box, as if some Masked Magician is fixing things here? Only if the choice was wrong by fluke, the treasure would be found in the remaining box.
In other words, there is no reason why the probability of two boxes with one having the treasure should get invalidated at all, just because someone accidentally peeped into one of the two empty boxes and showed the content. At least I can't find a reason that favors the probability of 2/3. The situation looks like completely reset case: it's all new two boxes of which one has treasure. What's the probability trick then?