Maths Fallacies & Parodoxes

1. The Barber's Paradox
Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men who do not shave themselves.
Under this scenario, we can ask the following question: Does the barber shave himself?
Asking this, however, we discover that the situation presented is in fact impossible:
If the barber does not shave himself, he must abide by the rule and shave himself.
If he does shave himself, according to the rule he will not shave himself.

2. The liar's Paradox
A man says that he is lying. Is what he says true or false?
Known to the ancients as the pseudomenon, the liar paradox encompasses paradoxical statements such as "This sentence is false." or "The next sentence is false. The previous sentence is true." These statements are paradoxical because there is no way to assign them a consistent truth value.

3. The Birthday Paradox (Not exactly a paradox, but answer is quite counter-intuitive) (http://www.efgh.com/math/birthday.htm)
A favorite problem in elementary probability and statistics courses is the Birthday Problem:
How large must N be so that there is > 50% chance that at least two of N randomly selected people have the same birthday (Same month and day, but not necessarily the same year)?

The answer is 23, which strikes most people as unreasonably small. For this reason, the problem is often called the Birthday Paradox. Some sharpies recommend betting, at even money, that there are duplicate birthdays among any group of 23 or more people. Presumably, there are some ill-informed who will accept the bet.


4. The Monty Hall Problem (Not exactly a paradox, but answer is counter-intuitive) Try it: http://math.ucsd.edu/~crypto/Monty/monty.html
The problem is also called the Monty Hall paradox, as it is a veridical paradox in that the solution is counterintuitive.

Here's the problem:
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?

Answer: Yes, switch your choice!
Because there is no way for the player to know which of the two unopened doors is the winning door, most people assume that each door has an equal probability and conclude that switching does not matter. In fact, in the usual interpretation of the problem the player should switch—doing so doubles the probability of winning the car, from 1/3 to 2/3.

A good info source on the problem: http://en.wikipedia.org/wiki/Monty_Hall_problem
Monte Hall problem - logical fallacy made by researcher in psychology experiment: http://www.nytimes.com/2008/04/08/science/08tier.html?em&
An excellent discussion forum with active & constructive discussions on the problem: https://nrich.maths.org/discus/messages/8577/7628.html?1077045704

An equivalant version using cards:
Suppose you have three cards:
a black card that is black on both sides,
a white card that is white on both sides, and
a mixed card that is black on one side and white on the other.
You put all of the cards in a hat, pull one out at random, and place it on a table. The side facing up is black. What are the odds that the other side is also black?

The answer is that the other side is black with probability 2/3. However, common intuition suggests a probability of 1/2.
In a survey of 53 Psychology freshmen taking an introductory probability course, 35 incorrectly responded 1/2; only 3 students correctly responded 2/3.
(Received from Hong Pin, 2008 July)