Life

Pascal’s Mugging

Remember Pascal’s Wager? It was an argument for the existence of God based on the idea of a payoff matrix, which is heavily dominated by the case in which God exists. So, the conclusion was to believe in God without seeking evidence. Something similar to Pascal’s wager but doesn’t require infinite rewards is Pascal’s mugging, a concept made by Eliezer Yudkowsky and later elaborated by Nick Bostrom.

Pascal was walking down the street. He was stopped by a shady-looking man asking for his wallet. It was an attempt to mug but without having any arms to threaten the victim. Knowing he has no gun to threaten Pascal, the mugger offers a deal. “Give me your wallet and I will give back double the money tomorrow.”

A utilitarian who trusts the expected value theory can make the following calculations and make a decision:
Imagine I have $10 in my wallet, and the minimum probability expected from the mugger keeping his promise is 50% for a break-even value.
This is because the expected value = – 10 + 0.5 x 20 = 0. Since the shady-looking man is not convincing enough to have such a high chance of repaying, I decide not to hand over my wallet.

Hearing the answer, the mugger increases the deal to 10x. That means if Pascal thinks the mugger has a 1/10 chance of paying 10x, he can hand over the wallet. The answer again was negative. The mugger increases the payment to 1000, 10000, and a million. What would Pascal do?

Now, Pascal is in a dilemma. On the one hand, he knows that the probability that the mugger will pay a million dollars is close to zero, and therefore, he must not hand over the wallet. However, as a rational utilitarian, he can’t ignore the fact that the calculations give a profit if the payback is 1 million dollars.

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Winning Russian Roulette

Ana and Becky want to play a safer version of Russian Roulette. The game starts with a coin toss. Whoever wins the toss puts a single round in a six-shot toy revolver, spins the cylinder, places the muzzle against a target, and pulls the trigger. If the loaded chamber aligns with the barrel, the weapon fires, and the player wins.

Given the rules of the game, how important is the toss?

We will evaluate the probability of each player – the one who wins (W) the toss and the one who loses (L).

Let’s write down the scenarios where the toss winner (W) wins the contest.
1) W wins in the first round
2) W doesn’t win in the first, survives the second (L’s chance) and wins the third.
3) W doesn’t win in the first, survives the second (L’s chance), doesn’t win the third, survives the fourth, and wins the fifth.
…

The total probability of W winning is the sum of all individual probabilities.

1) chance of winning the first round = 1/6
2) chance of winning the third round = chance of not winning the first x chance of not losing the second x chance of winning the third = (5/6)x(5/6)x(1/6) = (1/6)x(5/6)2
3) chance of winning the fifth = (1/6)x(5/6)4
Overall probability = 1/6 + (5/6)x(5/6)x(1/6) = (1/6)x(5/6)2 + (1/6)x(5/6)4 + (1/6)x(5/6)6 + … = 0.545 = 54.5%

On the other hand, the person who lost the toss has a total probability of 45.5% of winning the game. So, a 9% advantage is given by the toss.

Surviving The Deadliest 2-Player Game: Vsauce2

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People v. Collins v. Statistics

People v. Collins was a 1968 trial in the Supreme Court of California that reversed the convictions of Janet and Malcolm Collins by a jury in Los Angeles of second-degree robbery. The events that led to the original conviction were as follows:

On June 18, 1964, Mrs. Juanita Brooks was walking home along an alley in San Pedro, City of Los Angeles. She was suddenly pushed to the ground, and she saw a young woman running, wearing “something dark.” She had hair “between a dark blond and a light blond,”. After the incident, Mrs Brooks discovered that her purse, containing between $35 and $40, was missing.

At about the same time, John Bass, who lived on the street at the end of the alley, saw a woman run out of the alley and enter a yellow automobile driven by a black male wearing a moustache and beard.

The prosecutor (of the jury trial) brought a statistician to prove the crime using the laws of probability. The expert hypothesised the following chances and made his calculations.

Characteristic Probability
1Yellow automobile1/10
2Man with moustache 1/4
3Girl with poleytail 1/10
4Girl with blondehair Girl with blonde hair
5Interracial couple in a car1/10
6Interracial couple in car1/1000

The profession used the product rule (the AND rule) of probability to prove the case. Multiplying all the probabilities, he concluded that the chance for any couple to possess the characteristics of the defendants is 1/12,000,000! Note that such multiplication is only possible if the individual probabilities are independent.

But the statistician (and the jury) convenient ignored a few things:
1) The validity of the probabilities. Those were his ‘inventions’ without any support from data.
2) The events (characteristics) were not independent from each other. More likely than not, the person with a beard has a moustache.
3) Once, a blond girl and a black man with a beard were counted, talking about the low probability of an interracial couple in a car is wrong. Think about it—the probability of an interracial couple, given one is a blond girl and another is a bearded black man, must be close to 1.

References

People v. Collins: Justia US Law
The Worst Math Ever Used In Court: Vsauce2
A Conversation About Collins: William B. Fairley; Frederick Mosteller

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Pilot Hit

Data from a television production company suggests that 10% of their shows are blockbuster hits, 15% are moderate success, 50% do break even, and 25% lose money. Production managers select new shows based on how they fare in pilot episodes. The company has seen 95% of the blockbusters, 70% of moderate, 60% of breakeven and 20% of losers receive positive feedback.

Given the background,
1) How likely is a new pilot to get positive feedback?
2) What is the probability that a new series will be a blockbuster if the pilot gets positive feedback?

The first step is to list down all the marginal probabilities as given in the background.

Pilot OutcomeTotal
PositiveNegative
Huge Success0.10
Moderate0.15
Break Even0.50
Loser0.25
Total1.0

The next step is to estimate the joint probabilities of pilot success in each category.
95% of blockbusters get positive feedback = 0.95 x 0.1 = 0.095.
Let’s fill the respective cells with joint probabilities.

Pilot OutcomeTotal
PositiveNegative
Huge Success0.0950.0050.10
Moderate0.1050.0450.15
Break Even0.300.200.50
Loser0.050.200.25
Total0.550.451.0

The rest is straightforward.
The answer to the first question: the chance of positive feedback = sum of all probabilities under positive = 0.55 or 55%.
The second quesiton is P(success|positive) = 0.095/0.55 = 0.17 = 17%

Pilot Outcome
P(Positive)P(success|Positive)
Huge Success0.0950.17
Moderate0.1050.19
Break Even0.300.55
Loser0.050.09
Total0.551.0

Reference

Basic probability: zedstatistics

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The Advanced One

Which is the most advanced form of life?

Before we reach the answer, we must know that evolution is not a linear process (like a ladder), unlike what Aristotle thought, but a branched (like a tree). Consider the family of Apes. The apes (Hominoidea) branches into two families – the ‘great apes’ and the ‘lesser apes’.

The lesser apes contain a bunch of gibbons. The great apes are further divided into two – Homininae and orangutans.

Does this picture mean the orangutans and homininae are higher than the gibbons? No, and the scheme can also be drawn in the following way.

Homininae goes to gorillas and hominini. In other words, the gorillas and the hominini have a common ancestor. Hominini then goes to pan (chimpanzees and bonobos) and humans.

You may conclude that an advanced species goes from left to right. That is also not true. The picture can also be like this.

All these living lineages have the same amount of time to evolve; therefore, they are equal!

Reference

Ape: Wiki

Understanding Evolution: Berkeley

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Genetic Drift

We have seen natural selection as one mechanism of evolution. To clarify the whole process, this way of evolution involves two steps: 1) a random (accidental) modification of genes (mutation), followed by 2) a proliferation of a certain kind because it somehow fits well with its environment (natural selection).

But natural selection is not the only mechanism for evolution. Another means of evolution is genetic drift. In such cases, there could be no features of the genetically different versions of the species that have advantages or disadvantages from the environment. Still, random fluctuation causes one of the gene types (allele) to reduce its frequency. This also suggests that this feature is especially significant in small, isolated populations.

To give a (silly) example to distinguish the two types of evolution, we have seen the story of A moth named Biston betularia earlier. The survival probability of the black-coloured moths was higher in the industrialised (coal-polluted!) England. This is an example of natural selection. Now, think about a small population of 20 moths – 15 white and five yellows in a bright, normal neighbourhood. Someone accidentally stepped over the group, perishing seven whites and all (5) yellows. We know neither species had a special trait to withstand the boots. Yet, in the end, only eight whites were the only survivors. They then increased to a large group of white moths.

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Value of a Lottery

A lottery has the following three prizes and sells 2 million tickets
1) one bumper prize of 1 million dollars.
2) 100 first prizes of 10,000 dollars each.
3) 10,000 consolation prizes of 1 dollar each.

If one ticket costs 2 dollars, should you buy the ticket?

The answer to this question depends on two aspects.
A) The difference between the expected value and cost of a single ticket.
B) The risk appetite of the buyer.

Expected value of a ticket

EV = P(1,000,000) x 1,000,000 + P(10,000) x 10,000 + P(1) x 1
where P(1,000,000) is the probability of winning a million dollar = 1 / 2,000,000

EV = [1/2,000,000] x 1,000,000 + [100/2,000,000] x 10,000 + [10,000/2,000,000] x 1
= 1.005

The cost of a ticket ($2) is higher than the expected value of winning ($1). A risk-averse or a risk-neutral person would avoid it.

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Conditional Probabilities

Based on the following data, what is the probability of a person making no error in her tax returns without support from a tax advisor?

50% of individuals get help from a tax advisor to file their returns. The probability of an individual making an error in the tax return is 25%. The chance of the person making an error, given a tax advisor is helping, is 10%.

Let P(M|A) be the probability of not making the error, given an advisor is not helping.
Based on the Conjunction Rule,
P(M & A) = P(A) x P(M|A)
P(M|A) = P(M & A) / P(A)

P(A) = probability of no advisor
P(M & A) = joint probability of no error AND no advisor.

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Decision Quality

Sound decisions are core to achieving good outcomes. Decision quality (DQ) is the quality of a decision at the point it is made, regardless of its outcome. A decision framework should meet six requirements to reach DQ.

Appropriate frame: is about solving the right problem using the right people. The decision must have clarity of purpose, scope, boundaries and a conscious perspective.

Create doable alternatives: Give good choices within the frame. It will involve creativity, doability, breadth and completeness.

Relevant, reliable information: It may come from data and judgment. The issue with decision-making is that it is forward-looking, and all we have is data from the past. That means the data must be reliable and should describe the underlying uncertainties and biases.  

Clear values and tradeoffs: Focus on value creation and transparency of value matrix and tradeoffs.

Sound reasoning: Use the information for each alternative and get the one with the greatest value.

Commitment to action: During the decision process, it’s important to get the right people and resolve any conflicts. Quality is defined by the support across stakeholders and a team that is ready to take action.

Reference

An Introduction to Decision Quality: Strategic Decisions Group

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Cognitive Dissonance

What happens when a new idea or information contradicts your values or ideas? The mind feels the conflict and will resolve the issue by taking action until new ideas are consistent with the older ones. This lack of psychological agreement between two concepts is cognitive dissonance.

Consider an environmental activist who understands human-made. CO2 emissions and global warming when it comes to air travel. How will she react? She may do one or a mix of the following:
1) Reject air travel, at least where alternatives are available.
2) Change the belief in climate change that
2a) The whole theory is a scam
2b) Ignore as if nothing has happened
3) Get into a justification mindset that
3a) Others are also doing the same
3b) My travel is vital to justify a bit of emissions
3c) I did not cause the global warming in the first place

Cognitive dissonance: Wiki
Cognitive Dissonance: Sprouts

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