Drunkard's walk

I have been reading Leonard Mlodinov's book "Drunkard's Walk" which describes how randomness rules our lives. Here are a few examples of cognitive biases described in the book.

1. Pascal's Triangle - Has applications wherever you need to know the number of ways in which you can choose some number of objects from a collection that has an equal or greater number. Each number in the triangle is claculated by adding the numbers to the left and right of it in the previous (above) line.

2. Monty Hall Problem. Suppose the contestants on a game show are given the choice of three doors: Behind one door is a car (something valuable); behind the others, goats (less valuable). After a contestant picks a door, the host, who knows what's behind all the doors, opens one of the unchosen door, which reveals a goat. He then says to the contestant, "Do you want to switch to the other unopened door?" Is it to the contestant's advantage to make the switch?

Let us assume the initial guess, Lucky Guess scenario, was correct. The probability of it initially is 1/3 (since behind only one of the three doors is the valuable object). On the contrary, the probability of the initial guess being wrong, Wrong Guess scenario, is 2/3. Now, if the contestant's initial guess was wrong, which is twice as likely as he was right, with one of the two doors in the Wrong Guess scenario being opened and found to reveal a goat, it is guaranteed that the car will be behind the other door. So the choice for the contestant is to decide between the two scenarios. Since the Wrong Guess scenario is twice as likely as his initial Lucky Guess scenario, it makes logical sense to switch his choice.

3. Regression to the mean, an extraordinary event (or a major deviation from the mean) is most likely to be followed by a more ordinary/average event. Daniel Kahneman, while lecturing to a group of Israeli air force flight instructors at the Hebrew University, found that his students felt that punishing/scolding pilots for their mistakes was very effective, whereas rewarding them was counter-productive.

All the pilots had a certain ability to fly, and while their skills were incrementally improving due to the training, it would be impossible to notice massive/spectacular improvements from one manouver to the next. So it would be natural that if a pilot made an exceptionally good manouver - one far above his normal level of performance - then the odds would be that he would perform more closer to his norm, or a worser level, the next day. Similarly, a pilot who performed very badly today, is most likely to perform closer to his norm, or big improvement, tommorrow! The flight instructors' penalties and rewards hardly made a difference. Kahneman's contention is that rewarding positive behaviour is more effective than punishing mistakes.