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”.

Comments

  1. Ramiah KumarMarch 18, 2019 at 6:01 PM

    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.

    About 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?

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