Life

Cavaliers AND LeBron AND Playoff

We have seen one extreme of the AND rule of probability, where people forget to realise how the conjunction makes events rarer. A well-known case is Linda’s problem. Here is the pictorial representation of the AND rule, which combines three events. 

The shaded region shows the joint probability of A, B, and C. As the number of events increases, the ‘common area’ shrinks. There is another extreme case of conjunction fallacy, typically used by journalists. Read the following title that appeared in CBS Sports. 

Cavaliers win first playoff series without LeBron James since 1993 by taking Game 7 over Magic

The writer has combined Cleaveland’s playoff entry, the first-round victory, and Lebron’s absence, making it a ‘rare’ sensational event.  

1. Is this the first playoff entry? No, this young Cavaliers team has been playing well. They were also in the playoff last season (2022–23) but lost against the Nicks.
2. So this must be the first series win (ever)? No, they have recently made four consecutive NBA final appearances (2014-15, 2015-16, 2016-17, 2017-18). Note that these are not just one series victory; we are talking about championship finals—four times!
3. But this happened after a long time? No, the team won the first rounds in 2007-08, 2008-09 and 2009-10.
4. Then, how can I make it a rare event? Find things in common and start subtracting them. Win, Lebron, Decade, the list goes on.

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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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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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Screen Time and Happiness

The effect of screen time on mental and social well-being is a subject of great concern in child development studies. The common knowledge in the field revolves around the “dispalcement hypothesis”, which says that the harm is directly proportional to the exposure.

Przybylski and Weinstein published a study on this topic in Psychological Science in 2017. The research analysed data collected from 120,115 English adolescents. Mental well-being (the dependent variable) was estimated using the Warwick-Edinburgh Mental Well-Being Scale (WEMWBS ). The WEMWBS is a 14-item scale, each answered on a 1 to 5 scale, ranging from “none of the time” to “all the time.” The fourteen items in WEMWBS are:

1I’ve been feeling optimistic about the future
2I’ve been feeling useful
3I’ve been feeling relaxed
4I’ve been feeling interested in other people
5I’ve had energy to spare
6I’ve been dealing with problems
7I’ve been thinking clearly
8I’ve been feeling good about myself
9I’ve been feeling close to other people
10I’ve been feeling confident
11I’ve been able to make up my own mind about things
12I’ve been feeling love
13I’ve been interested in new things
14I’ve been feeling cheerful

The study results

I must say, the authors were not alarmists in their conclusions. The study showed a non-linear relationship between screen time and mental well-being. Well-being increased a bit with screen time but later declined. Yet, the plots were in the following form (see the original paper in the reference for the exact graph).

A casual look at the graph shows a steady decline in mental well-being as the screen time increases from 2 hours onwards. Until you notice the scale of the Y-axis!

In a 14-item survey with a 1-5 range in scale, the overall score must range from 14 (min) to 70 (max). Instead, In the present plot, the scale was from 40 to 50, thus visually exaggerating the impact. Had it been plotted following the (unwritten) rules of visualisation, it would have looked like this:

To conclude

Screen time impacts the mental well-being of adolescents. It increases a bit, followed by a decline. The magnitude of the decrease (from 0 screen time to 7 hr) is about 3 points on a 14-70 point scale.

References

Andrew K. Przybylski and Netta Weinstein, A Large-Scale Test of the Goldilocks Hypothesis: Quantifying the Relations between Digital-Screen Use and the Mental Well-Being of Adolescents, Psychological Science, 2017, Vol. 28(2) 204–215.
Joshua Marmara, Daniel Zarate, Jeremy Vassallo, Rhiannon Patten, and Vasileios Stavropoulos, Warwick Edinburgh Mental Well-Being Scale (WEMWBS): measurement invariance across genders and item response theory examination, BMC Psychol. 2022; 10: 31.

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Flight Accidents

YearAccidents
197624
197725
197831
197931
198022
198121
198226
198320
198416
198522

We assume that flight accidents are random and independent. This implies that the likelihood function (the nature of the phenomenon) is likely to follow a Poisson distribution. Let Y be the number of events occurring within the time interval.

Y|\theta = Pois(\theta)

Theta is the (unknown) parameter of interest, and y is the data (total of 10 observations). We will use Bayes’ theorem to estimate the posterior distribution p(theta|data) from a prior, p(theta). As we established long ago, we select gamma distribution for the prior (conjugate pair of Poisson).

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