The Drunkard's Walk, How Randomnes Rules Out Lives, by Leonard Mlodinow

[chin]

The Drunkard's Walk, How Randomnes Rules Out Lives, by Leonard Mlodinow

Another very interesting book. I got it this afternoon and read about half already.

The main line of the book follows the development of theory of randomness & probability (initially observed by gamblers, of course.) It's not dry history at all, but very captive narrative mixing with lots of interesting information, quiz and other bits and pieces.

(I am sure there are lots of reviews on the Internet for more details. I am just trying to write down part of the stories that interest me or touch my thought.)

[chin]

In the opening chapter, a good part was to illustrate how our life was affected by random events. I am quoting the following, hoping one day my kids will see the wisdom of it, and that the lesson make a lasting impression in their minds.

(page 10-11)
... That's why successful people in every field are almost universally members of a certain set - the set of people who don't give up.

A lots of what happens to us ... is as much the result of random factors as the result of skill, preparedness, and hardwork... That's not to say that ability doesn't matter - it is one of the factors that increase the chances of success - but the connection between actions and results is not as direct as we might like to believe.

Random events affecting one life are what some may call luck. As one grow older, the perception of what's luck changes too.

I once believed there is no such thing as lucky - that whatever good thing I haves, I deserve it because I worked hard and was smart enough. It take time to appreciate the random events that happen upon us, such that we can seize the opportunity and create good things out of it.

On the other hand, I have seen my share of people who are trying to read too much in random events, and becoming superstitious instead of taking events as they are.

[chin]

In Chapter 3 about sample space, and Gerolamo Cardano's attempt to solve a gambler's problem through relative simply probability concepts in 1500s, there is an interesting problem, supposedly called the Monty Hall problem.

(page 43)
Suppose the contestants on a game show are given the choice of three doors: Behind one door is a car; behind the others, goats. After a contestant picks a door, the host, who knows what's behind all the doors, opens one of the unchosen doors, 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?

What's your answer? and why?
Please try without lookup reference on the Internet or book.

[chin]

In the next chapter dealing with paths and possible combinations, there is a short mentioning of the birthday problem.

My friends from high school would remember that in our F4 (?) class, we had about 40 students and 3 of us have the same birthday! What's the odds of that?

Our math teacher explained the solution, and the chance wasn't as remote as I thought. But now I have forgotten the solution. Perhaps someone can remind me...

[chin]

In the same chapter, the author talked about Blaise Pascal's life and his contribution to solve the combinatory problem by the now famous Pascal's Triangle.

Toward the end of Pascal's life, he completely transformed himself to a religious man, and "finally blasted off from planet Sanity." (Page 75) One of the products during this period was what's called Pascal's Wager, where he argued "the pros and cons of one's duty to God as if he were calculating mathematically the wisdom of a wager." (Page 76)

In another other book I read many years ago, the Pascal's Wager was said to be the "rational" arguement that has driven people en mass to Christianity. The Pascal Wager basically argued from the Expected Value angle. The key is that 1/2 of infinity is still infinity.

I never believed in his arguement but was not able to counter the idea in terms of expected value. Come to think of it now, the problem of his arguement is that not only the probability of God (as definited in the religious context) exists or not is not 50/50, the payoff is not infinity and the downside of believing not as trivial as "the sacrifices of piety." Or expressed as

EV = Prob * Positive Payoff + ( 1 - Prob) * Negative Payoff

Consider when Positive Payoff is not infinite, Negative Payoff is not trivial, & Prob is small enough...

[Some wise man said never to discuss religion or politics with your friends. In any case, the above can only serve as theoritical discussion. I learned many years ago that religion is emotional (taking a leap of faith), not rational (as in proving one way or the other.)]

[chin]

Before I forget, Benford's Law & Zeno's Paradox are all very interesting. Maybe I will comment more later.

(I heard of the Zeno's Paradox or something similar before, but did not realize it's usefulness in explaining calculus.  )

[kido]

In the next chapter dealing with paths and possible combinations, there is a short mentioning of the birthday problem.

My friends from high school would remember that in our F4 (?) class, we had about 40 students and 3 of us have the same birthday! What's the odds of that?

Our math teacher explained the solution, and the chance wasn't as remote as I thought. But now I have forgotten the solution. Perhaps someone can remind me...

Really?  Who are the 3 guys?

Let's exercise our maths.  Consider a case of pre-fixed 3 people.  Given a day-of-year number[1..365], the prob. of another one has the same bday is 1/365, likewise, the prob. of a third guy having the same bday is 1/(365^2).  

Assume our class size was 40. There were 40C3 ways to choose 3 people... i.e. 9880 ways. At the end 40C3 / (365^2) = 7.4%

Anyone can verify it for me?

[hangchoi]

They are Chin, Yan and Lam Wing Kei....all 29/10

[chin]

Really?  Who are the 3 guys?

Let's exercise our maths.  Consider a case of pre-fixed 3 people.  Given a day-of-year number[1..365], the prob. of another one has the same bday is 1/365, likewise, the prob. of a third guy having the same bday is 1/(365^2).  

Assume our class size was 40. There were 40C3 ways to choose 3 people... i.e. 9880 ways. At the end 40C3 / (365^2) = 7.4%

Anyone can verify it for me?

Just happened that we had similar discussion in the office on the same problem. We figured that the logic goes like this:

a. pick some one who has a birthday, any birthday. The chance of this is 100%.
b. randomly pick another person, and the chance of having the same birthday would be 1/365.
c. so by now, we established that for any PARTICULAR pair, the chance of having same birthday is a * b or 1/365.
d. in a group of n people, the # of permutations of pairs is nC2, thus the chance of having two persons having same birthday is nC2/365.

Example 1: I often heard that 23 people is enough to have a GOOD chance that a pair will have the same birthday. Using d above, 23C2/365=253/365 or about 2/3 chance.

Example 2: We have about 40 classmates in F5, 3 having same birthday. Using the same logic about, the chance should be 40C3/(365*365)=7.4% which is exactly what Kido had!  (How clever we were...  )

[wongyan]

Let's say,

you play Sik-Bo in Casino.

After four SMALLs in a roll, engineering/scientists mind-set will always say the next game is 1/2 Big, 1/2 Small.  While some other people will start to believe the dice are biased and the chance of SMALL is higher on the next game.  So everything may come to an art.  I believe Chin will agree also.  No matter how accurate you model your wagering system, the assumptions behind are all set by human.

Let's get back to the three doors dillemma.  I will look into the host's eyes and say "Yes..............Maybe" and see if any clue from his eyes or his facial expression can tell.  Besides, I may also knock on the door and see if the goat bleats.  (Probability tells no difference)