Lesser known probability puzzles with counter-intuitive solutions

Question

I'm tutoring an introductory course in statistics next semester so I would like to give my students some fun counter-intuitive problems to think about.

There are some famous problems like the Monty-Hall problem or the Birthday Paradox, but I feel that many students would most likely already know those.

What are some lesser known probability problems with paradoxical or counter-intuitive seeming solutions?

Cheers!

Answer 1

Here's an activity that is fun to have students make predictions (guesses) first, and then simulate:

Imagine giving $100$ gifts to $100$ children so that each gift is equally likely to go to any of the $100$ children, independently of what happens with any other gifts (i.e., equivalent to putting $100$ names in a hat and doing $100$ draws with replacement from the hat).

You can ask questions such as: How many children will not get any gifts? How many children will get exactly one gift? How many gifts will the 'luckiest' child get?

You can also extend to asking...If there is an unlimited number of gifts, how many gifts will be distributed until each child has gotten at least one gift?

Usually the predictions are off by quite a bit from what actually occurs.

Answer 2

My favourite Statistics paradox is Simpson's paradox, which has consequences in everyday life. It is quite easy to put it as a problem.

Answer 3

Another really counterintuitive paradox is Parrondo's paradox, where a combination of losing strategies becomes a winning strategy. It took me a while to grasp it :-)

Answer 4

$66$ points in $100$ shots.

In the above problem, all the scores are equally likely is a bit hard to believe.

Answer 5

In a unit circle, where should you draw a unit line segment, to maximize the probability that it will be intersected by a random chord? The chord is drawn by connecting two uniformly random points on the circle.

The correct answer is C. In my experience, the most popular choice is A, followed by B, then C, but their actual probabilities are $\frac{1}{\pi^2}\int_0^\pi \arccos{\left(\frac{3}{\sqrt{25-16\cos^2{x}}}\right)}dx\approx 0.209, \frac{1}{4}=0.250, \frac{5}{18}\approx 0.278$, respectively. C gives the maximum probability of all possible locations of the line segment. See here for more information.

Answer 6

Consider a hemisphere with horizontal base, with the base on the bottom. Point $A$ is the top point, and point $B$ is a uniformly randomly point on the surface of the hemisphere. $L$ is the shortest distance from the centre of the base to the line segment $AB$. So $\sqrt2/2\le L\le1$.

Question: Is $f_L(l)$, the probability density function of $L$, increasing or decreasing?

Surprisingly, it is increasing.

For an explanation, see this answer, "Step 1", and the comments section.