In a game involving a fair coin, each toss is an independent event with a fixed 50% probability of landing on heads or tails. If the coin lands on heads 10 times in a row, a person committing the Gambler's Fallacy might incorrectly conclude that the next flip is highly likely to be tails because a tails outcome is "due." This reasoning is mathematically flawed because the coin has no memory of previous flips.

Coin Toss Example of the Gambler's Fallacy

The Gambler's Fallacy is the incorrect belief that the probability of an independent random event is affected by prior occurrences. Individuals committing this fallacy falsely assume that if an event occurs more frequently than expected over a given period, it is statistically less likely to happen in the future, or vice versa, despite the events being completely independent.

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A few of the common statistical paradoxes are:"

- Monty Hall's paradox 
- Berkson's paradox
- Simpson's paradox

Common statistical paradoxes

Monty Hall is a famous paradox from a reader's letter to "Ask Marilyn" column in Parade magazine in 1990.

The problem is defined as: suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

While the instinctive reaction is usually to think that the probabilities of either door are the same, the conditional probabilities that the car is behind door 1 and door 2 are 1/3 and 2/3 respectively. Thus the game show participant should always switch their choice, given the option. 


Monty Hall’s Paradox

This a subcategory of a collider bias (Collider is the selection of the data). When two factors individually are usually not severe enough to cause some effect but the combination of the two is. Usually both factors are not thought to be correlated with each other in real life settings but in a certain condition they are. This seems to occur because the resulting effect is being adjusted for by not considering the situations where both factors do not occur. This association is caused based on the conditional collider of the environment the data is selected. 

Berkson's Paradox

When groups of data show one particular trend, but this trend is reversed when the groups are combined together. (A/B) > (a/b) & (C/D) > (c/d) but (A+C)/(B+D) < (a+c)/(b+d)

Simpson's Paradox

Low Birth weight babies have a significantly higher mortality rate than others. The smoking mothers are more likely to have low birth weight babies than non-smoking mothers. Thus, people inferred that the child mortality rate should be higher among children of smoking mothers. However, the observation runs contrary to our expectation. Low birth weight babies of smoking mothers have a lower child mortality than low birth weight babies of non-smokers.

Although smoking may contribute to low weight and be harmful for babies, other causes like genetic abnormalities (birth defect) are much more harmful than smoking. Those underweight babies would weight normal if their mother were non-smoker. But they still have a lower mortality rate than babies who have more severe medical reasons that made them born underweight.

We can explain this paradox with a causal diagram.  In this case, birth weight is the collider, a variable which is a common effect of two other variables, smoking and U(other causes like birth defects). Conditioning on the collider will open a back-door path that goes smoking -> birth weight <-U(birth defects) -> Infant death. The collider bias is so large that even makes smoking seems beneficial. 


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