[Book] Chances and Successes -- The Drunkard's Walk Wrap-up

Although regularity can be found in social data, the future of a particular individual is impossible to predict due to

• There are infinitely many possibilities,
• Minor change can lead to a significant change in outcomes (i.e., the Butterfly Effect), and 
• People are irrational. They sometimes act against their best interest. 

For a particular action, we all owe more to the chance (i.e., luck) than what most people realize.

[Book] Past-Present Asymmetry

 “If the future is really chaotic and unpredictable, why after the event has occurred does it seem as if we should have been able to foresee them?”

It is easy to reconstruct (or explain) a present from the past. But it s virtually impossible to predict the future from the present. To reconstruct the present, we can look at the past and make up the reason which leads to the present. If we use the same approach to predict (or extrapolate) the future, before long we will end up with infinitely many possibilities. And, even if you can magically pinpoint the future, the Butterfly Effect can mess things up. A minor event from the present to future can change the future entirely. This is to say “the future is unpredictable”.

Source: The Drunkard's Walk: How Randomness Rules Our Lives by Leonard Mlodinow, Chapter 10 [Read the Book Review] [Read the Previous Part] [Read the Next Part].

[Book] Butterfly Effect

No, we are not talking about the movie by Ashton Kutcher. We are going to talk about the Butterfly Effect which is a concept in chaos theory, which says

A flap of butterfly's wings in Brazil can set out a tornado in Texas. 

More generally, few insignificant random events can cause a big impact. This phenomena occur a lot in our daily life. But perhaps, it is referred to as fate, rather than the Butterfly Effect. For example, had Bruce Willis not gone to at attend an Olympic event in 1984, he might have not been a star today, or had you not gone to study abroad, you might have not found the love of your life.

Source: The Drunkard's Walk: How Randomness Rules Our Lives by Leonard Mlodinow, Chapter 10 [Read the Book Review] [Read the Previous Part] [Read the Next Part].

[Book] Hot hand fallacy

 Hot hand fallacy states that if performing well in last consecutive trials, a person might perform well in the next trials. In fact, the streak (of well-performance) may just be because of pure chance. If this is the case, the results in the few consecutive trials infer nothing about the next trials.


Hot hand fallacy is perceived in various fields such as sport or business. We usually judge people (e.g., basketball players or companies) by statistics. But we usually misled by a hot hand fallacy. When a good player makes several baskets consecutively, we tend to think that his hand is “hot”, and that his teammates should give him the ball. So if an average player makes several baskets consecutively, does is mean that his is better than a good player? No, because if he is better, his statistics would have been better. The fact that he makes a lot of baskets consecutively implies that he might miss a lot of baskets too. We just don’t know when it will happen.

[Book] People, Perception, Perfection, and Randomness

 Perfection requires perfect perception. It is hard to achieve perfection, since the data we perceives is rarely perfect. So how do we determine whether what we perceive is true. In a more technical term, how do we accept or reject the hypothesis based on what we perceived? In this respect, statisticians resort to "Significant Testing" in order to make decision based on observations.

"Human perception is very narrow. There is only about 1 degree of visual angle around the retina center which has high resolution. Outside this region, the resolution drops off sharply. Therefore, we tend to move our eyes a lot to compensate for the narrow visual area."

[Book] The Dawn of Statistics

Statistics began as early as 1066, when William the conqueror, the Duke of Normandy, conquered and became the King of England. He would like to find out what exactly he did conquer and how much tax he can collect. So he sent out two groups of inspectors to men to do some survey on land and livestock. At that time, people believed that it is the God’s will to let people live or die. So, a survey of people as well as how they born and die were forbidden. As time changes, people changes. In later time, people began to believe that study of population is not against the God’s will, but is the way to understand him better. In the 16th century, the London Bill of Mortality was drafted. It was the first attempts to record people’s birth and death.

"Statistics can provide insights into the system from which the statistics are derived."

Source: The Drunkard's Walk: How Randomness Rules Our Lives by Leonard Mlodinow, Chapter 8 [Read the Book Review] [Read the Previous Part] [Read the Next Part].

[Book] Measurement and the Law of Errors

Ancient Experiment: Science or Art
In the old time, scientists perform experiment, and make the best intuitive guess of what the result should be. And, if they were to repeat the experiment, they could get totally different results. They just lacked a standard for measurement.
In other words, they use feeling rather than scientific method. Is this science? Or, is it art? “Any variation within a margin of error should be ignored.”

[Book] False Positives and Positive Fallacies

Conditional Probability

We learned earlier that more details can lead to more restriction and reduce the corresponding probability. More specifically, the probability of event A is always greater than the probability of events A and B.

But details do not always reduce the probability. It can increase the probability by shrinking the sample space. We can be more sure that a specific outcome will occur if we can cross off some of the sample space. This is known as “Bayes’ Theorem”.

Probability, Ratio, and the Law of Large Numbers

Probability and Ratio

You probably know about probability and ratio. Suppose I put 70 of black balls and 30 of white balls in an urn. General statements would be

• The Ratio of black balls (when comparing to all the balls in the urn) is 7:10
• If I draw a ball from the urn, the Probability that I would draw a black ball would be 7 in 10

Gods and Expectation

Blaise Pascal

Blaise Pascal is known as the person who invented the Pascal's Triangle, whose elements are the number of ways to arrange N objects in M places.

But, the story that drew my interest is that he is also the inventor of expectation. And, even more interesting is how he came up with the concept of expectation. Here we go....

The Birthday Problem

Given that everyone’s birthday is absolutely random, how many people
must be in the room such that there is more than even chance (at least
50%) that

• Another person would have the same birthday as you (ANS: 365/2 =
  183 persons)
• Two persons would have the same birthday (ANS: 23 persons)

The difference is that you fix the date in the former. But you let the date vary in the latter. It is obvious that you would need less people in the latter than in the former. But the surprising result is that it is a lot less.

Sample Space and Gambling

The Dawn of Sample Sapce

Gerolamo Cardano was an physician, a gambler, and a mathematician. Early in his career, he discovered the concept of sample space, and tried to publish the concept in a book named “The book on Games of Chance”. The manuscript was rejected.

He did not try to published it again. Instead, he used the concept to make a fortune out of gambling. The book was published after his death in 1663. Here is the quote from the book.

“The possible outcomes of a random process can be thought of as a point in the space.”

This space of all possible outcomes is later known to as “sample space”.

The Interpretation of Randomness

How much do you know about randomness? What does it mean by a random process? Well, a lot of people would say that a random process is a process whose outcomes cannot be predicted. Clean and simple, and nice for a lot of people.

But this definition is quite vague to mathematician. Mathematicians have something for preciseness. When you say a circle is round, they would ask how round, and you would not know how to reply to them. But if you ask a mathematician how round is a circle, the answer would be 3.14 x R x R where R is the radius.

Likewise, the above definition for a random process is not precise enough. For mathematician, there are at least two interpretation for a random process:

(i) Frequency interpretation: A process is said to be able to generate random numbers when the observed outcome conform to the underlying probability.

(ii) Subjective interpretation: Observed outcomes are said to be random if they cannot be predicted.

The Greek, the Roman, and the Mathematics

The Greek and Their Mathematics

Most greek Mathematics are mainly about geometry. They do not believe in probability since

• They believe that chance is governed purely by Gods.
• Their number system is very difficult to work with. Algebra and arithematics had not existed in their time. They don’t even have the number zero nor fraction.

Availability Bias

A Trick Question You Might Want To Try With Your Friends

Let me ask you a question. Among English 6-letter words, which of the following is more probable?

• Find a word whose fifth letter is ‘n’.
• Find a word ending with ‘ing’.

Regression towards the Mean

A result which falls far from the mean tends to be closer to the mean during the next attempt. Failing something consecutively does not mean that the next attempt will not succeed. Attempts and successes are not linearly related.

Future is unpredictable, but there is one good news. If you are good (i.e., the mean is good enough), you will finally succeed (i.e., reach the mean). Do not give in, even if you have failed (i.e., fall off the mean) consecutively.

“Genius does not guarantee success, but it is seductive to assume that success must come from genius.”

The logic behind the “regression towards the mean” theorem is as follows. Suppose your average test score is 50. Now, let’s take a test. Suppose that you’ve got 40. If your next test scores less than 40, which is further away from the mean, your average won’t be 50. It has to be lower than 50 which contradicts to what we assume at the beginning.

Source: The Drunkard's Walk: How Randomness Rules Our Lives by Leonard Mlodinow, Chapter 1 [Read the Book Review] [Read the Previous Part] [Read the Next Part].

Are we a better guesser than a mouse?

It is compelling to answer ‘yes’. Unfortunately, the answer is ‘not always’. To see why, let’s setup an experiment where red and green cards are shown to a test subject. The job of the subject is to observe a sequence of cards and to guess the color of the next cards. Suppose that we use a certain sequence where 75% of the cards are green and 25% are red cards. The subject might choose to use one of the following 2 strategies for guessing:

(i) Probability based guessing: Observe and compute probability of cards being green and red. Then always choose the color with higher probability. In the above example, the subject will always choose green. If we let the subject keeps guessing, the correct probability will approach 75%. But it can never be better than 75%. This is what most non-human animals such as mouses would do.