Counterintuitive examples in probability

I want to teach a short course in probability and I am looking for some counter-intuitive examples in Probability. The results seem to be obviously false but they true or vice versa.

I already found some things. For example these two videos:

• Penney's game
• How to win a guessing game

In addition, I have found some weird examples of random walks. For example this amazing theorem:

For a simple random walk, the mean number of visits to point $b$ before returning to the origin is equal to $1$ for every $b \neq 0$.

I have also found some advanced examples such as Do longer games favor the stronger player?

Could you please do me a favor and share some other examples of such problems? It's very exciting to read yours...

Solution 1:

The most famous counter-intuitive probability theory example is the Monty Hall Problem

• In a game show, there are three doors behind which there are a car and two goats. However, which door conceals which is unknown to you, the player.
• Your aim is to select the door behind which the car is. So, you go and stand in front of a door of your choice.
• At this point, regardless of which door you selected, the game show host chooses and opens one of the remaining two doors. If you chose the door with the car, the host selects one of the two remaining doors at random (with equal probability) and opens that door. If you chose a door with a goat, the host selects and opens the other door with a goat.
• You are given the option of standing where you are and switching to the other closed door.

Does switching to the other door increase your chances of winning? Or does it not matter?

The answer is that it does matter whether or not you switch. This is initially counter-intuitive for someone seeing this problem for the first time.

• If a family has two children, at least one of which is a daughter, what is the probability that both of them are daughters?
• If a family has two children, the elder of which is a daughter, what is the probability that both of them are the daughters?

A beginner in probability would expect the answers to both these questions to be the same, which they are not.

Math with Bad Drawings explains this paradox with a great story as a part of a seven-post series in Probability Theory

Nontransitive Dice

Let persons P, Q, R have three distinct dice.

If it is the case that P is more likely to win over Q, and Q is more likely to win over R, is it the case that P is likely to win over R?

The answer, strangely, is no. One such dice configuration is $(\left \{2,2,4,4,9,9 \right\},\left \{ 1,1,6,6,8,8\right \},\left \{ 3,3,5,5,7,7 \right \})$

Sleeping Beauty Paradox

(This is related to philosophy/epistemology and is more related to subjective probability/beliefs than objective interpretations of it.)

Today is Sunday. Sleeping Beauty drinks a powerful sleeping potion and falls asleep.

Her attendant tosses a fair coin and records the result.

• The coin lands in Heads. Beauty is awakened only on Monday and interviewed. Her memory is erased and she is again put back to sleep.
• The coin lands in Tails. Beauty is awakened and interviewed on Monday. Her memory is erased and she's put back to sleep again. On Tuesday, she is once again awaken, interviewed and finally put back to sleep.

In essence, the awakenings on Mondays and Tuesdays are indistinguishable to her.

The most important question she's asked in the interviews is

What is your credence (degree of belief) that the coin landed in heads?

Given that Sleeping Beauty is epistemologically rational and is aware of all the rules of the experiment on Sunday, what should be her answer?

This problem seems simple on the surface but there are both arguments for the answer $\frac{1}{2}$ and $\frac{1}{3}$ and there is no common consensus among modern epistemologists on this one.

Ellsberg Paradox

Consider the following situation:

In an urn, you have 90 balls of 3 colors: red, blue and yellow. 30 balls are known to be red. All the other balls are either blue or yellow.

There are two lotteries:

• Lottery A: A random ball is chosen. You win a prize if the ball is red.
• Lottery B: A random ball is chosen. You win a prize if the ball is blue.

Question: In which lottery would you want to participate?

• Lottery X: A random ball is chosen. You win a prize if the ball is either red or yellow.
• Lottery Y: A random ball is chosen. You win a prize if the ball is either blue or yellow.

Question: In which lottery would you want to participate?

If you are an average person, you'd choose Lottery A over Lottery B and Lottery Y over Lottery X.

However, it can be shown that there is no way to assign probabilities in a way that make this look rational. One way to deal with this is to extend the concept of probability to that of imprecise probabilities.

Solution 2:

Birthday Problem

For me this was the first example of how counter intuitive the real world probability problems are due to the inherent underestimation/overestimation involved in mental maps for permutation and combination (which is an inverse multiplication problem in general), which form the basis for probability calculation. The question is:

How many people should be in a room so that the probability of at least two people sharing the same birthday, is at least as high as the probability of getting heads in a toss of an unbiased coin (i.e., $0.5$).

This is a good problem for students to hone their skills in estimating the permutations and combinations, the base for computation of a priori probability.

I still feel the number of persons for the answer to be surreal and hard to believe! (The real answer is $23$).

Pupils should at this juncture be told about quick and dirty mental maps for permutations and combinations calculations and should be encouraged to inculcate a habit of mental computations, which will help them in forming intuition about probability. It will also serve them well in taking to the other higher level problems such as the Monty Hall problem or conditional probability problems mentioned above, such as:

$0.5\%$ of the total population out of a population of $10$ million is supposed to be affected by a strange disease. A test has been developed for that disease and it has a truth ratio of $99\%$ (i.e., its true $99\%$ of the times). A random person from the population is selected and is found to be tested positive for that disease. What is the real probability of that person suffering from the strange disease. The real answer here is approximately $33\%$.

Here strange disease can be replaced by any real world problems (such as HIV patients or a successful trading / betting strategy or number of terrorists in a country) and this example can be used to give students a feel, why in such cases (HIV patients or so) there are bound to be many false positives (as no real world tests, I believe for such cases are $99\%$ true) and how popular opinions are wrong in such cases most of the times.

This should be the starting point for introducing some of the work of Daniel Kahneman and Amos Tversky as no probability course in modern times can be complete without giving pupils a sense of how fragile one's intuitions and estimates are in estimating probabilities and uncertainties and how to deal with them. $20\%$ of the course should be devoted to this aspect and it can be one of the final real world projects of students.