Decision Making

The Necktie Paradox

Two friends, Andy and Boris, are making a bet about their neckties, which their spouses gifted them (so they don’t know the price). The bet goes like this: each will call the spouse, and whoever has the cheaper tie wins and gets the other person’s more expensive tie. 

Andy reasons that he had a 50/50 chance of winning the bet. If he loses, he loses the value of his tie. If he wins, he wins more than the value of his tie. In other words, there is a 50% probability he loses x and a 50% chance he gets more than x. So, he must wager. 

Boris also thinks the same. Obviously, this is impossible, where both men have the advantage in this game. 

There is a logical error in their reasoning. The person’s argument of losing x and gaining more than x suggests he thinks he will lose a less expensive tie and get a more expensive tie, which is not correct. Let $20 be the price of a tie and $40 the other. Since each doesn’t know the price at the time of the bet, there are four possibilities with a 25% chance each. 

Andy has a $40 tie, and Boris has a $20 tie – Andy’s gain: – $40
Andy has a $20 tie, and Boris has a $40 tie – Andy’s gain: + $40
Andy has a $40 tie, and Boris has a $40 tie – Andy’s gain: 0
Andy has a $20 tie, and Boris has a $20 tie – Andy’s gain: 0

The expected value is: 0.25 x -40 + 0.25 x 40 + 0.25 x 0 + 0.25 x 0 = 0. The same goes for Boris. 

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Bread Sales

A bread salesman makes an average of 20 cakes; what is the chance he sells an even number of cakes? 

Let’s assume the sales follow a Poisson distribution. 

Reference

Fifty Challenging Problems In Probability: Frederick Mosteller

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Mother vs Girlfriend

Adam finishes work at random times between 3 PM and 5 PM. His mother lives uptown, and his girlfriend lives downtown. After work, he goes to the metro station, catches the first train in either direction and has dinner with the person who lives on the side he reaches. His mother complains that his son came only two times in the last twenty working days, whereas Adam thought he was fair, with either party getting 50-50 chances. What’s really happening? 

Suppose the metro on each side runs every 10 minutes, making it six times an hour: 3:00, 3:10, 3:20, etc. Let D represent the metro to downtown and U to uptown at x minutes after D. 

Since the mother saw him only twice in twenty days, the probability of Adam catching the uptown metro is 2/20 = 1/10, which must be equal to x/10. This implies x = 1. So the metro that goes uptown reaches its stop at 3:01, 3:11, 3:21, etc.

Reference

Fifty Challenging Problems In Probability: Frederick Mosteller

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Bayesian Persuasion

Persuasion is the act of a person (a.k.a. the sender) to convince another (the receiver) to decide in favour of the sender. Suppose the receiver is a judge and the sender is the prosecutor. The prosecutor aims to make the judge convict 100% of the defendants. But the judge knows that only a third of the defendants are guilty. Can the prosecutor persuade the judge to get more than 33% of the decisions in her favour? If the judge is rational, what should be the prosecutor’s strategy?  

Suppose the prosecutor has the research report and the knowledge about the truth. She can follow the following three strategies.

Strategy 1: Always guilty

The prosecutor reports that the defendant is guilty 100% of the time, irrespective of what happened. In this process, the prosecutor loses credibility, and the judge resorts to the prior probability of a person being guilty, which is 33%. The result? Always acquit the defendant. The prosecutor’s incentive is 0. 

Strategy 2: Full information

The prosecutor keeps it simple – report what the research finds. It makes her credibility 100%, and the judge will follow the report, convicting 33% and acquiring 66%. The prosecutor’s incentive is 0.33. 

Strategy 3: Noisy information

Here, when the research suggests the defendant is innocent, report that the defendant is guilty slightly less than 50% of the time and innocent the rest of the time. Let this fraction be 3/7 for guilty and 4/7 for innocent. 

From the judge’s perspective, if she sees an ‘innocent’ report from the prosecutor, she will acquit the defendant. The proportion of time this will happen is (2/3) x (4/7) or 40%. Remember, 2/3 of the defendants are innocent! On the other hand, she will apply the Bayes’ rule if she sees a guilty report. The probability that the defendant is guilty, given the prosecutor provided a guilty report, P(g|G-R), is

P(g|G-R) = P(G-R|g) x P(g) / [P(G-R|g) x P(g) + P(G-R|i) x P(i)]
= 1 x (1/3) /[1 x (1/3) + (3/7) (2/3)]
= (1/3)/(13/21) = 0.54

The judge will convict the defendant since the probability is > 50%. So, the overall conviction rate is 100 – 40 = 60%. The prosecutor’s incentive is 0.6. 

Conclusion

So, persuasion is the act of exploiting the sender’s information edge to influence the receiver’s decision-making. As long as the sender mixes up the flow of information to the judge, she can maximise the decisions in her favour, in this case, from 33% to 60%. 

Emir Kamenica and Matthew Gentzkow, American Economic Review 101 (October 2011): 2590–2615

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The Rating Problem

Here is the rating summary of a product,
Good – 40%
Average – 10%
Poor – 50%
Looking at the product, how do you know which view represents the actual quality of the product?

Can we conclude that the probability of the product being good equals 0.4, average 0.1, and poor 0.5? Although that is what we want from the rating system, we must realise that these may not represent the absolute or marginal probability of quality but the conditional probability, e.g., the probability of good a given person has rated. In other words

P(Good|Rated) = 0.4
P(Average|Rated) = 0.1
P(Poor|Rated) = 0.5

From this information, we can estimate the actual probabilities, P(Good), P(Average) and P(Poor) using Bayes’ theorem. 

P(Good|Rated) = P(Rated|Good) x P(Good) / P(Rated)
P(Average|Rated) = P(Rated|Average) x P(Average) / P(Rated)
P(Poor|Rated) = P(Rated|Poor) x P(Poor) / P(Rated)

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P-Hacking

P-hacking is an often malicious practice in which the analysis is chosen based on what makes the p-value significant. Before going into detail, let’s recall the definition of p-value. It is the probability that an effect is seen purely by chance. In other words, if we chose 5% as the critical p-value to reject (or fail to reject) a null hypothesis, 1 in 20 tests will result in a spectacular finding even when there was none. 

So what happens if the researcher carries out several tests and reports only the one with the ‘shock value’ without mentioning the context of the other non-significant tests? It becomes an example of a p-hacking. 

References

P-Hacking: Crash Course Statistics: CrashCourse
Data dredging: Wiki
The method that can “prove” almost anything: TED-Ed

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Confusion of the Inverse

What is the safest place to be if you drive a car, closer to home or far away?

Take this statistic: 77.1 per cent of accidents happen 10 miles from drivers’ homes. You can do a Google search on this topic and read several reasons for this observation, ranging from overconfidence to distraction. So, you conclude that driving closer to home is dangerous. 

However, the above statistics are useless if you seek a safe place to drive. Because what you wanted was the probability of an accident, given that you are near or far from home, say P(accident|closer to home). And what you got instead was the probability that you are closer to home, given you have an accident P(closer to home|accident). Look at the two following scenarios. Note that P(closer to home|accident) = 77% in both cases.

Scenario 1: More drive closer to home

Home Away
Accident7723100
No
Accident		10002001200

Here, out of the 1300 people, 1077 drive closer to home.
P(accident|closer to home) = 77/1077 =0.07
P(accident|far from home) = 23/223 = 0.10
Home is safer. 

Scenario 2: More drive far from home

Home Away
Accident7723100
No
Accident		10005001500

Here, out of the 1600 people, 1077 drive closer to home.
P(accident|closer to home) = 77/1077 =0.07
P(accident|far from home) = 23/523 = 0.04
Home is worse 

This is known as the confusion of the inverse, which is a common misinterpretation of conditional probability. The statistics only selected the people who had been in accidents. Not convinced? What will you conclude from the following? Of the few tens of millions of people who died in motor accidents in the last 50 years, only 19 people died during space travel. Does it make space safer to travel than on earth? 

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The Bayesian Runner

Becky loves running 100-meter races. The run timing for girls her age follows a normal distribution with a mean of 15 seconds and a standard deviation of 1s. The cut-off time to get into the school team is 13.5 seconds. If Becky is on the school running team in 100 meters, what is the probability that she runs below 13 seconds?

Without any other information, we can use the given probability distribution to determine the chance of Becky running under 13 seconds.

pnormGC(13,region="below", mean=15,sd=1.,  graph=TRUE)

Since we know she is in the school team, we can update the probability as per Bayes’ theorem. Let’s use the general formula of Bayes’ theorem here:

\\ P(13|T) = \frac{P(T|13)*P(13)}{P(T)}

The first term in the numerator, P(T|13) = 1 (the probability of someone in the team with a cut-off of 13.5 s, given her timing is less than 13s). We already know the second term, P(13), 0.0228.

The denominator, P(T), is the probability of getting on the team, which is nothing but the chance of running under 13.5 seconds. That is, 

pnormGC(13.5,region="below", mean=15,sd=1.,graph=TRUE)

Substituting P(T) = 0.0668 in the Bayes equation,

P(13|T) = 0.0228/0.0668 = 0.34 or 34%

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Begging the Question

A person new to religion may have questions about the credibility of its belief system. For example, how do I know the way God operates? A person with a Christian belief system will tell you it’s written in the Bible. But how do I know the Bible is telling the truth? I did a search, and one of the results on the validity of the scriptures was the King James Bible online. The first one was:

“2 Timothy 3:16 – All scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness:” 

If you are happy with the answer, like millions of people, you have committed the fallacy of begging the question. Begging the question is a complicated way of describing a circular reasoning, where an argument’s premise assumes the truth of the conclusion.

An argument has three parts:
1. Claim or conclusion – presented upfront
2. Support or evidence – presented to back the claims
3. Warrant or major premise – hidden in the argument but bridges the support to the claim.

Consider the following argument. 

In the fallacy of circular argument, the claim (Bible is telling the truth) takes the evidence (verses from the book of Timothy) from the same book to prove the claim.

Data & Statistics on Autism Spectrum Disorder: CDC

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Liar Puzzle

Annie says Becky is lying
Becky says Carol is lying
Carol says both Annie and Becky are lying

Who is lying, and who is telling the truth?

  1. If Carol is telling the truth, then Annie (and Becky) is lying. In that case, Becky is not lying (opposite of what Annie, the ‘liar’ said). This is a contradiction. Therefore, Carol is lying.
  2. Since Carol is lying, the person who identified that correctly, Becky, must be telling the truth
  3. If Becky is telling the truth, the first statement is incorrect. Therefore, the person who made the first statement, Annie, is also lying.

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