What is Simpson's paradox?

Question

Jon Wayland · Answer

During a faculty meeting, a group of 9th grade teachers decided they needed to further understand what the optimal duration of study is for students to achieve satisfactory results. So, they decided to gather the approximate number of hours students were studying, and then compare to the student’s test scores.

Mr. Simpson convinced the faculty that more data means better results, and so all of the teachers integrated their cross-course data for the analysis.

The results were astounding. To everyone’s confusion, the less a student studied, the higher they tend to score on tests.

In fact, the coefficient associated with this correlation was -0.7981, a strongly negative relationship.

Should they be encouraging their students to study less? How in the world could data be backing up such a claim? Surely something was missing.

After discussing the results, the teachers agreed they should consult the school’s statistician, Mrs. Paradox. After Mr. Simpson explained to Mrs. Paradox what they had found in their results, Mrs. Paradox suggested they analyze each course’s data individually.

So, they went ahead and analyzed Phys. Ed. and proceeded to have their minds blown.

A correlation of 0.6353! How in the statistical universe was this even possible?

Mrs. Paradox then explained this as Simpson’s Paradox, a statistical phenomenon where a seemingly strong relationship reverses or disappears when introduced to a third confounding variable.

She convinced Mr. Simpson to plot all of the data once again, but then color-code each course separately to distinguish them from one another.

After doing so, Mr. Simpson and the 9th grade faculty concluded that the relationship was indeed positive, and that the more hours a student studied, the higher the grade tends to be.

Including the course of study in the analysis completely reversed the relationship.

R Code for this example:

[code]# Load the tidyverse
library(tidyverse)

# Generating correlated data with mvrnorm() from the MASS library
library(MASS)

# Sample Means
mu %3C- c(20,4)

# Define our covariance matrix, and specify the covariance relationship (i.e. 0.7 in this case)
Sigma %3C- matrix(.7, nrow=2, ncol=2) + diag(2)*.3

# create both variables with 100 samples
vars %3C- mvrnorm(n=100, mu=mu, Sigma=Sigma)

# Examine the data and the correlation
head(vars)
cor(vars)

# Plot the variables
plot(vars[,1],vars[,2])

# Create a function for generating 2 correlated variables given variable means
corVars%3C-function(m1,m2,confVar){
  mu %3C- c(m1,m2)
  Sigma %3C- matrix(.7, nrow=2, ncol=2) + diag(2)*.5
  vars %3C- mvrnorm(n=100, mu=mu, Sigma=Sigma)
  Var1%3C-vars[,1]
  Var2%3C-vars[,2]
  df%3C-as.data.frame(cbind(Var1 = Var1,Var2 = Var2,Var3 = confVar))
  df$Var1%3C-as.numeric(as.character(df$Var1))
  df$Var2%3C-as.numeric(as.character(df$Var2))
  df
}

# Re-running for multiple sets and combining into a single dataframe df
d1 %3C- corVars(m1 = 20, m2 = 82, confVar = "Algebra")
d2 %3C- corVars(m1 = 18, m2 = 84, confVar = "English")
d3 %3C- corVars(m1 = 16, m2 = 86, confVar = "Social Studies")
d4 %3C- corVars(m1 = 14, m2 = 88, confVar = "Art")
d5 %3C- corVars(m1 = 12, m2 = 90, confVar = "Physical Education")

# Create the aggregate data
df%3C-rbind(d1,d2,d3,d4,d5)

# Grade & Study Time Plot
df %%3E%
  ggplot(aes(x = Var1, y = Var2/100)) +
  geom_jitter(aes(size = 13), alpha = 0.55, shape = 21, fill = "darkgray", color = "black") +
  scale_y_continuous(name = "Final Percentage", labels = percent)+
  scale_x_continuous(name = "Approximate Hours for Preparation")+
  guides(size = FALSE) +
  ggtitle("Impact of Studying on Final Grades")+
  theme(plot.title = element_text(hjust = 0.5))+
  theme_bw()

# Grade & Study Time Correlation
cor(df$Var1, df$Var2)

# PhysEd Plot
df %%3E% 
  filter(Var3 == 'Physical Education') %%3E%
  ggplot(aes(x = Var1, y = Var2/100)) +
  geom_jitter(aes(size = 13), alpha = 0.55, shape = 21, fill = "darkgray", color = "black") +
  scale_y_continuous(name = "Final Percentage", labels = percent)+
  scale_x_continuous(name = "Approximate Hours for Preparation")+
  guides(size = FALSE) +
  ggtitle("Impact of Studying on Final Grades (Physical Education Only)")+
  theme(plot.title = element_text(hjust = 0.5))+
  theme_bw()

# PhysEd Correlation
cor(df$Var1[df$Var3 == 'Physical Education'], df$Var2[df$Var3 == 'Physical Education'])

# Confounding plot
df %%3E%
  ggplot(aes(x = Var1, y = Var2/100)) +
  geom_jitter(aes(size = 1, fill = Var3), alpha = 0.25, shape = 21) +
  guides(fill = guide_legend(title = "Course Class", override.aes = list(size = 5)),
         size = FALSE) +
  scale_y_continuous(name = "Testing Results", labels = percent)+
  scale_x_continuous(name = "Approximate Hours for Preparation")+
  ggtitle("Impact of Studying on Final Grades")+
  theme(plot.title = element_text(hjust = 0.5))+
  theme_bw()
[/code]

Mitch Johnson · Answer

Simpson’s Paradox says that a broad trend, when broken down to individual components, may be manifest to be the reverse of the apparent.

Bob and Sue are weight-training coaches. Who’s a more successful coach?

Bob’s lifters bench an average of 191 lbs.

Sue’s lifters bench an average 159.1 lbs.

Is Bob more successful? No. Bob coaches 10 guys who bench 200 lbs each and he coaches one girl who benches 100 lbs.

To calculate the average, we take (10*200 lbs + 1* 100 lbs)/11 people.

Sue coaches 10 girls who each bench 150 lbs and one guy who benches 250.

To calculate the average, we take (10*150 lbs + 1* 250 lbs)/11 people.

So if just average everything, it looks like Bob is a more successful coach. If you look at the individual breakdowns, it’s clear that Sue is the better coach. Or at least it’s clear that her women can outlift Bob’s woman, and her one man can outlift Bob’s men.

The broader lesson is that when you start averaging things across dissimilar groups, you run into errors.

Aaron Brown · Answer

It’s not a paradox. It’s a common situation in data analysis that researchers should be alert for. It’s common for a trend in a combined data set to have the opposite sign when the data are broken into subpopulations, or results are adjusted for another variable.

For example, I was working on a sex discrimination case against United Airlines in the 1970s. At the time, all the pilots were men, and all the cabin attendant's were women. The pilots generally went to the military after high school to get their training. The stewardesses usually had some college or were graduates. The pilots made four times the money that the stewardesses did.

If you graphed income versus education, you saw a downward slope—high-school-graduate pilots made a lot, college-educated stewardesses made little.

But if you looked at pilots separately, the ones with more education made more money. The same was true of stewardesses.

There’s no paradox, no contradiction. The issue is that putting the two populations together without adjusting for sex (or for job type, which meant the same thing at the time) produced a misleading, non-causal trend.

ChatGPT · Answer

Simpson's paradox is a phenomenon in statistics where a trend that appears in several different groups of data disappears or reverses when these groups are combined. This paradox highlights how aggregated data can lead to misleading conclusions, as the relationship between variables may differ when examined at a more granular level.
Example:
Imagine two treatments for a disease, A and B. In separate groups of patients (e.g., men and women), treatment A appears to be more effective than treatment B. However, when the data from both groups is combined, treatment B shows better overall effectiveness. This can happen if one group has a significantly larger number of patients or if the groups differ in ways that affect the outcomes, such as age or severity of illness.
Key Points:
 * Aggregation Issues: Aggregating data can obscure important differences between groups.
 * Confounding Variables: A confounding variable can influence the results, leading to different conclusions at different levels of analysis.
 * Importance of Context: It emphasizes the need for careful analysis and consideration of how data is grouped in statistical studies.
Simpson's paradox serves as a cautionary tale in data analysis, reminding researchers and analysts to carefully consider how they present and interpret their findings.

Mike Tamada · Answer

There have been some good examples given, here’s another real-world one.

There’ve been some years when, if we look at the SAT scores of all students who took the test, the mean score fell (compared to the previous year).

But when we look at students by ethnicity, the mean score of white students rose. And the mean score of black students rose. And Asians, and everybody else (including “unknown/declined to state their ethnicity”).

How can every single group have had higher scores, but the overall total population had lower scores?

Answer: Simpson’s Paradox. The mix of students changed, with more students from lower-scoring groups taking the test. Each group had higher scores — but there were fewer students from the high-scoring groups and more from the low-scoring ones.

Simpson’s Paradox tells us that what’s true for an overall pattern might be reversed when we look at sub-groups. Or another way to put it, if you account for ethnicity (or add it as an explanatory variable to your model), the pattern can change.

Sanjay Kumar · Answer

Simpson’s paradox is a statistical phenomenon where the same trend occurs in different groups of data points but when all the groups are combined, the trend gets reversed.

For example,

Let’s say that we need to compare the performance of 2 international cricket batsmen across a period of time.

Consider that the following table shows the batting average of 2 batsmen – Sachin Tendulkar and James from 2010 – 2015.

Clearly, James has a higher batting average than Sachin in all the years. Does it mean James is a better batsman than Sachin?

Hence, Sachin %3C James

Let’s look at the following table also-

Sachin has scored 26708 runs in 438 matches from 2010 -2015 that makes a total batting average of around 60.97

James has scored 28387 runs in 471 matches from 2010 -2015 that makes a total batting average of around 60.26

So, Sachin has a higher batting average than James in total.

Hence, Sachin %3E James

Now, we cannot consider James as a better batsman than Sachin. This contradicts the inference from the first table. This is nothing but Simpson's paradox.

According to standard definition- “Simpson's paradox is a phenomenon in statistics, in which a trend appears in several different groups of data but disappears or reverses when these groups are combined.”

Here, when we examined the batting average of Sachin and James year by year, James appeared to be a better batsman but when we aggregated their overall performance together then Sachin appeared as a better batsman than James.

Usually, this scenario happens in many use cases due to the presence of a hidden bias in the data. In the above-mentioned example, there is a drastic difference in the number of matches played by Sachin and James in the years 2010 and 2013 which made a way to contradictory inferences.

Similarly, in any problem, there is a possibility of an unnoticed bias that can lead to Simpson's paradox.

Arithmetically, when (a1/A1) %3C (a2/A2) and (b1/B1) %3C (b2/B2) we tend to think (a1+a2)/(A1+A2) %3C (b1+b2)/(B1+B2)

But Simpson's paradox proves that this may not be true in all the cases and that’s what happened in the above example.

Let us understand the below scenario for better intuition-

This image shows the relationship between the happiness and age of a person according to a survey.

The below diagram clearly shows that the happiness of a person increases when he/she gets older ie age and happiness are positively correlated.

But if we split the age into multiple groups then we can see that -

This may be because of the reasons that-

* Happiness decreases from the age of 10 to age 20 (This may be because children become very happy when they join the high schools and gets new friends and teachers but eventually happiness goes down due to academic pressure).
 * Happiness is suddenly increasing after the age of 20 but again starts to decrease till the age 30 (This may be because people usually become very happy in early 20’s when they are bachelors and get a job but eventually happiness goes down due to work pressure and financial commitments).
 * Happiness is suddenly increasing after the age of 30 but again starts to decrease till the age of 40 (This may be because people become very happy once they get married in the early ’30s but eventually happiness goes down due to relationship issues).
 * Still, the overall happiness of any person increases when they are becoming more mature as they are getting older from childhood.
 * In each subgroup, there is a negative correlation between happiness and age.
Simpson’s paradox tells us to uncover the hidden details by discovering deep insights in the data before reaching a statistical conclusion. Otherwise, it may be resulting in wrong inferences.

This concept is praised and followed by many researchers who work on machine learning and artificial intelligence across the globe.

Hope it helped you !!!

Anonymous · Answer

This is one of the most interesting paradox. Almost every person must have encountered it once in his life. It adversely affects our decision making as you will see, so beware of it!

Let me explain it with a simple example:
Suppose you and I are captains of 2 football teams A and B.
Each of the team played 10 matches in 2 months.

In the first month,
Team A won 0 match out of the 3 it played. So your win % is 00.00%
Team B won 1 match out of the 7 it played. My win % is 14.20%

In the second month,
Team A won 5 matches out of 7 it played. So your win % is 71.40%
Team B won 3 match out of the 3 it played. My win % is 100.00%

So when the statisticians put out the numbers, here is what the report card looked like: 
Team    Captain    Month 1(win %)   Month 2(win %)
    A           You               00.00                        71.40
   B            Me                14.20                       100.00

And experts unanimously declare me as a much better captain than you(oh yeah!) and Team B much better than Team A.
But are they correct?
NO

Here is what the the report card of 2 months looks like
Team    Total matches played     Total matches won     Win(%)
    A                       10                                            5                           50
    B                       10                                            4                           40

You won more matches than I did and both the teams played equal number of matches. Surprising isn't it!? This paradox plays on our mind since the result is so counter-intuitive. 
One of the best-known real-life examples of Simpson's paradox occurred when the University of California, Berkeley was sued for bias against women who had applied for admission to graduate schools there. (Page on berkeley.edu [ http://www.cs.berkeley.edu/~daw/teaching/cs70-s08/notes/n21.pdf ])

The admission figures for the fall of 1973 showed that men applying were more likely than women to be admitted, and the difference was so large that it was unlikely to be due to chance.But when examining the individual departments, it appeared that no department was significantly biased against women. In fact, most departments had a "small but statistically significant bias in favor of women."!!

So beware the next time you judge someone on the basis of statistics!

Quique Dopico · Answer

A real life example.

In Spain, global mortality rate by cardiovascular diseases is higher in women than in men.

But each and every age-specific mortality rate by these diseases are higher in men than in women. So, a woman, at a certain age, has less risk of dying because of these deseases than a same age man, other things equal.

But , as there are many more women from 75–80 years on, where the majority of these deaths occur, the global mortality rate by sex indicates just the opposite.

Timothy Michael · Answer

Simpson’s paradox occurs when data subsets show trends that disappear or reverse when viewing the full data set.

Here is a simple example, borrowed from Wikipedia:

Justice had a higher batting average than Jeter in both 1995 and 1996. But when you combine the data from both years, Jeter had the higher average. How is this possible?

The key is the size of the data sets. Both of them had a better year in 1996 than 1995, but Jeter had a lot of at-bats in his good year and few in his bad year. Similarly, Justice had more at-bats his bad year, and relatively few his good year. Jeter’s good year gets weighed more heavily when the data is combined, as does Justices bad year, and Jeter winds up with the higher average.

This is just one way Simpson’s paradox can occur, it doesn’t have to be about sample sizes. It will usually some overlooked variable that isn’t being accounted for, either in the divided sets or the full data set.

Anonymous · Answer

As a cartographer and spatial analyst, this speaks to the problems of generalization. The characteristics of a group as a whole may be significantly different from the characteristics of included groups.

For example, a green area on a map shows a “forest,” but it omits significant information about the structure and composition of that forest area. One area might be mostly Douglas Fir and Hemlock, for example, while another area is Big Leaf Maple and Alder. For a more glaring example, the arid and relatively sparse forests of Nevada are shown with the same symbol as the dense forests of coastal Oregon.

And I’ve just barely started. There are some other good answers here, too. So, remember well the phrase mistakenly attributed to Mark Twain and others: lies, damned lies, and statistics.

Dan Tannehill · Answer

OK, I see some qualified folks have already given quality answers. So I’ll throw a wrench into the works and give a different “Simpson’s Paradox.” This is “Bart Simpson’s Paradox.”

https://youtu.be/w6TpSnkOu18?t=170
You’re damned if you do and damned if you don’t.

Huyen Nguyen · Answer

The Low Birth Weight paradox

Today everybody knows that smoking causes lung cancer. But it wasn’t so obvious in the 1950s and took nearly a decade to resolve this debate. Even after the smoking and cancer debate died down, a major puzzling paradox remained. In the mid 1960s, it was pointed out that a mother’s smoking seemed to have benefits for her newborn baby if the baby was born underweight. This so-called birth-weight paradox was a major challenge to the emerging medical consensus about the harmful effects of smoking and took more than 40 years to resolve.

In 1959, Jacob Yerushalmy, a biostatistician at the University of California at Berkeley and also an opponent of the smoking-causes-cancer hypothesis, launched a long-term public health study that collected data on more than 15,000 children in the San Francisco Bay Area. The data included information about mothers’ smoking habits, the birth weights and mortality rates of their babies in the first month of life.

It was well-known that low birth-weight (LBW) infants, defined as less than 2500g at birth, have a mortality rate more than twenty times higher than that of normal-birth-weight infants. Babies of smoking mothers weigh less at birth than babies of non-smokers. It would seem natural to assume this lower birth weight (LBW) would lead to poorer survival.

Yet what Yerushalmy found was surprising. The LBW babies of smoking mothers had a better survival rate than those of nonsmokers. It was as if the mother’s smoking had a protective effect!

Yerushalmy was not silly enough to make this absurd claim but he stated that there is no causal link between smoking and mortality. But this seems odd and counter-intuitive.

And it is just plain wrong. Modern epidemiologists believe that smoking does increase infant mortality. But how can the data be explained?

To make it more concrete, we can look at a more recent dataset of over three million children born in the US in 1991 presented in this paper Birth Weight “Paradox” Uncovered? [ https://academic.oup.com/aje/article/164/11/1115/61454 ] which observed the same thing that Yerushalmy did 50 years ago (btw, if you find this paper full of jargons and hard to read, I did too but bear with me I’ll make it clearer). Overall, infants born to smokers had higher risks of both LBW and mortality than infants born to nonsmokers.

But among LBW infants, the mortality rate was lower for infants born to smokers. You can see in this plot below that the switch happens around 2000g.

I can’t find the raw data for this paper so I try to simulate in this table:

While the overall mortality rate of the smoker-mother group (3+3)/(55+445) = 1.2% is greater than that of the non-smoker-mother group (10+11)/(150+2350) = 0.8%, you can see in the LBW group, infants of smoker-mothers have lower mortality rate (5% vs 7%).

Using causal diagrams, the authors finally uncovered the reason behind this paradox. Smoking may be harmful and contributes to LBW, but there are other more serious and harmful causes of LBW such as genetic defects or malnutrition. If the mother is a smoker, this ‘explains away’ the LBW and reduces the likelihood of a serious birth defect. If the mother is not a smoker, it is likely that the cause of the LBW is a serious birth defect which leads to higher mortality. The presence of these unmeasured risk factors (U) can induce a spurious association between smoking mortality when the analysis is split by birth weight.

Only the three arrows marked by the red circles in the diagram above are enough to create this paradox. This bias is introduced when the data is split on a variable (weight) that is affected by the exposure (smoking) and shares common causes (U) with the outcome (mortality).

It was lucky that the apparent explanation was so ridiculous (smoking is beneficial) that the bias was detected. There might be many other cases where this bias is undetected because it doesn’t conflict with theory.

Refs:

Birth Weight “Paradox” Uncovered? [ https://academic.oup.com/aje/article/164/11/1115/61454 ]

On the importance—and the unimportance— of birthweight [ https://academic.oup.com/ije/article/30/6/1233/651751 ]

The Book of Why: The New Science of Cause and Effect - Judea (University of California, Los Angeles) Pearl [ https://www.amazon.com.au/Book-Why-Science-Cause-Effect/dp/046509760X ]

Vanessa Lukas · Answer

Simpson's paradox is a statistical phenomenon where a trend or association that holds for different groups reverses when the groups are combined.

This occurs due to the differing rates of aggregation, which can lead to confusing conclusions.

One surprising aspect of Simpson's paradox is that it highlights the importance of considering the context in which data is analyzed.

For instance, a study on the effectiveness of a new medication might show a positive correlation between the medication and patient recovery rates when analyzed separately for different age groups, but a negative correlation when the data is combined.

Another interesting aspect is that Simpson's paradox can arise from the way we aggregate data.

Aggregation can introduce biases that are not present in the original data, and these biases can lead to wrong conclusions.

For example, in a study on the relationship between university tuition fees and graduation rates, the data might show a positive correlation when analyzed separately for different ethnic groups, but a negative correlation when the data is combined.

Simpson's paradox also has real-world implications, such as in healthcare, where it can lead to misdiagnosis or ineffective treatment.

In medical research, the paradox can arise when combining data from different studies, leading to conflicting conclusions.

Furthermore, Simpson's paradox is not unique to statistics; it can occur in any field where data is aggregated, including economics, social sciences, and business.

It highlights the need for a deeper understanding of the underlying mechanisms and the importance of considering the context in which data is analyzed.

In addition, Simpson's paradox can be mitigated by using more advanced statistical techniques, such as regression analysis, which can help account for the underlying structures in the data.

This emphasizes the importance of careful consideration of the statistical methods used in data analysis.

Andrei Kucharavy · Answer

Paradox:
Simpson's paradox occurs when your sample is composed of separate classes with different mean values of a statistical value. In this case, if the class distribution within sample changes between two measures, the trend observed on average might be opposed to the trend observed in each of the two classes.

Example: 
Let's say that a teacher has a class with 100 students, 10 among them come from a disadvantaged background and their average in year 1 is 80/100. All the other students are normal and have an average of 90/100. Thus the average of her class was at 89/100. Since disadvantaged student were feeling themselves well with that particular teacher they advised other disadvantaged students to sign in into the course and in the year 2 the same teacher had 50 students from a disadvantageous background and 50 normal students. Normal student average became 91 and disadvantaged ones - 81. However, the average of the class dropped to 86 and she gets a call from the dean asking her why did she become so bad at teaching.

Conclusion:
Always look for a meaningful split of your data into classes that might have different behavior. If not, you might obtain a correlation opposed to the real one. If your data don't look like a Gaussian, don't try to pretend it is a Gaussian and proceed anyway: instead try to split it into classes that look Gaussian.

Jeff Suzuki · Answer

The testing paradox (also known as the Prosecutor’s Fallacy):

You have a test that is 90% accurate (correctly identifying 90% of those with a condition, and 90% of those without the condition). You test someone, and the test result is positive. How likely are they to have the condition?

The correct answer is: It depends on how prevalent the condition is in the population. If the condition is only present in 10% of the population, then there’s only a 50% chance that the person actually has the condition.

The first time I encountered this problem was taking the Math GREs. I got the answer, and didn’t believe it, so I rechecked my work and got the same answer. I thought about it some more, and had to do something like the following to convince myself the answer was, in fact, correct:

https://youtu.be/9PXgX0j0Nkg?list=PLKXdxQAT3tCvuex_E1ZnQYaw897ELUSaI
That was in 19mumblemumble. Since then, testing mania has taken over, with drug tests being mandatory for a lot of things, from getting a job flipping burgers to receiving public assistance. Given the high likelihood of a false positive result, and the impossibility of proving you weren’t using drugs at the time of the test, the wisdom of using such tests is highly questionable.

Robert Rounthwaite · Answer

When you compare a population with labeled subpopulations with another population (or "the same" at a different time), it's extremely likely that the two populations will have different proportions of their subpopulations. This is the heart of Simpson's paradox.

It's easiest to understand if you think about the change happening with one population over time.

Let's take a very simple example. You've got girls and boys. The girls have, on average, longer hair than the boys do, and there is an average for the school somewhere in between. Now, a boy shows up with longer than average hair (for a boy) but shorter than the school average. Presto: The girls have the same length hair as before. The boys have, on average, longer hair. Even though no subgroup has shorter hair, the school average has gone down!

This is the way it goes with Simpson's paradox -- the groups have averages that go in one direction while the overall average goes in the other. Sometimes it's because members of the population leave or join, sometimes it's due to shifts in counts within the sub-groups. But it's always because the counts in subgroups differ between the two populations.

Read more here: What are some recent examples of Simpson's Paradox in the media? [ http://qr.ae/G5jLg ]

Pooja Chauhan · Answer

An example is False positive paradox [ https://en.wikipedia.org/wiki/False_positive_paradox ] : We would like to think that if we devise a Test, we should be able to measure its accuracy in a lab But the accuracy of a Test also depends on the characteristics of the population on which it is being applied. This is surely counter intuitive, but true nevertheless.

Suppose we invent a new Test which gives 95% correct diagnosis, and 5% false positives. We have tested it over large population samples. We apply it to a population of 100 people in which 40% people have the disease. The test works well. 
 * 40 People will be diagnosed correctly as having the disease. 
 * 3 People will be diagnosed as having the disease but incorrectly - The False Positive. 
 * 57 People will be diagnosed correctly as not having the disease.
We can calculate the accuracy of the Test as the probability of correct result. As 40 of the 43 people detected positive are truly positive, we can say the accuracy is about 93%, not far from the initial 95%. So far, the test is good.

The Paradox comes when the same test is applied to another population which has a much lower prior probability of having the disease, lower than the false positive rate. Let's apply the above test to a new population of 100 people in which only 1% of people have the disease. 
 * 1 person will be diagnosed correctly as having the disease. 
 * 4.95 People ~ 5 People will be diagnosed incorrectly as having the disease - The False Positive  
 * The remaining 94 are correctly diagnosed as not having the disease. 
If we look at the accuracy of the test now, only 1 person of 6 detected actually has the disease. Something that was 93% accurate is now only 16% accurate. Hence the Paradox.

Wikipedia has a nice segment applying this to counter terrorism measures:

%3E

%3E Terrorists are really rare. In a city of twenty million like New York, there might be one or two terrorists, maybe up to ten. 10/20,000,000 = 0.00005 percent, one twenty-thousandth of a percent.

That's pretty rare. Now, say you have software that can sift through all the bank-records, or toll-pass records, or public transit records, or phone-call records in the city and catch terrorists 99 percent of the time.

In a pool of twenty million people, a 99 percent accurate test will identify two hundred thousand people as being terrorists. But only ten of them are terrorists. To catch ten bad guys, you have to investigate two hundred thousand innocent people.

Aditya Patil · Answer

Potato paradox

This is not a logical; but a mathematical paradox. Interesting nonetheless.

The paradox has been described as:

Let’s say you have 100 kg potatoes, which are 99% water by weight.

Now, you leave them outside overnight to dehydrate until they're 98% water.

How much do they weigh now?

The answer is 50 kg.

How so?

When you first had 100 kg of potatoes, water was 99% by weight, it means that there is 99 kg of water, and 1 kg of solids.

It's a 1:99 ratio.

Now, when potatoes dehydrate causing water to decrease to 98%, then the solids will account for 2% of the weight.Right?

So the ratio will become 2:98 which will reduce to 1:49.Since the solids still weigh 1 kg, the water must weigh 49 kg.That makes total weight 50 kg.

Mind-boggling, isn’t it?

Nikunj Nathany · Answer

The ultimate paradox is the plot of the movie 'Predestination (2014)'...
or basically the short story - 'All you Zombies' written by Robert Heinlein.

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%3E SPOILERS AHEAD!

Its overall premise is as follows:
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In an orphanage lived a small, little girl.

She grows up, turns 18, meets a man, has sex with him and delivers a daughter.
But the father abducts his daughter one night and disappears from the girl's (now a woman) life.

The woman then decides to have a sex change operation.

She turns into a man and discovers time travel.

The man then goes back into the past, meets his younger self (who is still a woman for now) and has sex with her.

The woman delivers a daughter which he abducts and flees.

He then goes further 18 years back into the past and then leaves his baby girl on the doorstep of an orphanage.

No points for guessing who the little baby girl turns into.

%3E 
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This is the ultimate paradox that was formulated in the story.
A perfect and mind-tripping paradox where a human being has no ancestory at all, has no beginning in nature and has just come into existence out of absolutely nowhere!!

And the most boggling part:
If, as and when time travel is discovered in the future, this scenario will ACTUALLY, suddenly become plausible (assuming that the cause and effect relationship is still valid and true after the possibility of time travel is verified).

Sleep on this!

P.S: As an additional food for thought, just mull over the question that what will be his genetic code and who shall he look like?!

P.P.S: Bootstrap paradox [ http://en.wikipedia.org/wiki/Bootstrap_paradox ] ...link for the explanation of all these kinds of existential paradoxes.

Peter Hauge · Answer

An Infinity paradox:

A bouncing ball returns to 1/2 of its initial height; each bounce completes in 1/2 the time of the previous bounce. Eventually it comes to rest. Whenever it hits the ground its color alternates between blue and red.

If it starts out blue, what color is it at rest?

Ant on a rubber band

An ant crawls along a rubber band from one end toward the other at a constant rate of 1 mph (edit: with respect to the rubber band.) The other end is tied to a car that travels away from the ant at a constant rate of 10 mph. (The rubber band has the capacity of being stretched indefinitely without breaking.)

Even though the car’s speed is 10 times that of the ant, the ant eventually reaches the car.

The unexpected hanging

On Saturday a condemned man was sentenced to be hanged. The hanging will take place at noon on one of the seven days of next week, said the judge. You will not know the day of your execution until it is announced to you on the morning of the hanging.

After some thought the prisoner concluded that his execution could not be on the following Saturday, because on Friday afternoon he would be alive with only Saturday remaining as an execution day. So he would know of his execution prior to the announcement on Saturday morning. With Saturday thus ruled out, the same argument can be applied to Friday. Similarly, Thursday, Wednesday, Tuesday, Monday, and finally Sunday can be ruled out. The prisoner reasoned, quite logically, that the hanging therefore could not be carried out at all, without contradicting the judge’s condition of surprise. The prisoner was overjoyed.

On Wednesday, quite to the prisoner’s surprise, the hangman announced his execution.

Non-transitive coin tosses

If a coin is tossed three times, eight equally probable patterns obtain: HHH, HHT, HTH, HTT and the four patterns obtained by exchanging H and T.

A game can be construed where one player picks a pattern and the other player picks a different one. A coin is flipped repeatedly until the pattern picked by one of the players appears as a run, and that player then wins the game. For example if Player 1 chooses THT and Player 2 chooses HHT and the flips are THHHT, Player 2 wins the game.

Even though the patterns have equal likelihood for three flips, Player 2, who chooses after Player 1, can secure a winning probability that is at worst 2/3 (2 to 1 odds) and at best 7/8 (7 to 1 odds).

Edit: For example if Player 1 selects HHH, Player 2 selects THH. Player 1 wins only if HHH are the first three flips (1 chance in 8). If HHH appears anywhere else in the sequence of flips it is preceded by a T and Player 2 wins.