What are some interesting un-intuitive problems in probability aside from Monty Hall?

Question

Does anyone know of some interesting and almost strange problems in probability?

I know that probability is sometimes notorious for being mind-bending and un-intuitive! (Monty Hall is already an obvious one)

Answer 1

Probably one of the oldest, that still gets students to raise their eyebrows is Buffon's Needle. The probability a drop of a needle of length $1cm$ onto $1cm$ lined paper will cross a line is $\dfrac{2}{\pi}$. This gives a useful way to estimate $\pi$, too, if you've forgotten it and are trapped on a desert island with only twigs and pack of lined paper (run the experiment a few million times to estimate the probability, take the reciprocal and multiply by $2$).

Another one, which is pretty darn nonintuitive is how being given seemingly useless information can change the probability of an event. Example: If I flip two fair coins and I tell you that at least one of the two coins came up Heads, and ask you what the probability is that both of them are Heads, you may not jump to the correct answer right away (or maybe you will): $\dfrac{1}{3} \approx 0.333$. But here is the fun part. If I instead tell you that at least one of the coins is Heads and was manufactured on a Saturday, will you arrive at the correct answer of $\dfrac{13}{27} \approx 0.481$?

For information on this problem read: https://en.wikipedia.org/wiki/Boy_or_Girl_paradox#Information_about_the_child

Edit: Thought I'd provide some more information on the coin/Saturday thing, because of the comments. If we denote dates with Saturday=1, Sunday=2, etc. Then given the information that at least one is a Heads and was manufactured on a Saturday ("Heads 1"). then give dates to all possibilities of both coins, our reduced sample space is:

$$(Heads 1, Heads 1) \\ (Heads 1, Heads 2) \\ (Heads 1, Heads 3) \\ (Heads 1, Heads 4) \\ (Heads 1, Heads 5) \\ (Heads 1, Heads 6) \\ (Heads 1, Heads 7) \\ (Heads 1, Tails 1) \\ (Heads 1, Tails 2) \\ (Heads 1, Tails 3) \\ (Heads 1, Tails 4) \\ (Heads 1, Tails 5) \\ (Heads 1, Tails 6) \\ (Heads 1, Tails 7) \\ (Heads 2, Heads 1) \\ (Heads 3, Heads 1) \\ (Heads 4, Heads 1) \\ (Heads 5, Heads 1) \\ (Heads 6, Heads 1) \\ (Heads 7, Heads 1) \\ (Tails 1, Heads 1) \\ (Tails 2, Heads 1) \\ (Tails 3, Heads 1) \\ (Tails 4, Heads 1) \\ (Tails 5, Heads 1) \\ (Tails 6, Heads 1) \\ (Tails 7, Heads 1)$$

(see https://js.do/caffeinatedlogic/676775)

There are 27 in total, only 13 of which are both Heads. I hope that clears it up.

Answer 2

Shuffle a deck of cards (standard $52$-card bridge deck) and deal one card. What's the probability that it's an ace? The answer is $\frac1{13},$ nothing unintuitive about that.

Shuffle a deck of cards and deal one card after another until an ace is dealt, then deal one more card. What 's the probability that the last card is an ace? Intuitiveness is a matter of opinion; personally, I find it kind of unintuitive that the probability is still $\frac1{13}.$

(More generally, for $n\ge k\ge2,$ if balls are drawn one by one without replacement from an urn containing $n$ balls of which $k$ are black, then the probability that the first black ball is immediately followed by another black ball is equal to $k/n.$)

Answer 3

An interesting introductory probability problem is one of basic geometry. Say you have a circle with an inscribed equilateral triangle. The question is, if you were to randomly draw a chord on this circle, what is the probability that it will be greater than the side length of the triangle?

You will come to 2 (or possibly more) conclusions. Thus, this problem emphasizes the importance of being precise when stating a question in probability. Saying "randomly selecting a chord on the circle" is the ambiguous statement that allows one to arrive at several conclusions about the probability.

Answer 4

Simpson's paradox : mixing does not preserve correlation sign (the mixing of two distributions having positive correlation on a pair of components can produce a distribution with negative correlation). Perhaps you need to be familiar with contingency tables to better grasp why this is surprising.

Answer 5

there's the rare illness test - despite no symptoms, a man takes a test for an illness that occurs in one in a billion people. The test always detects the illness if it is present, and it delivers negative correctly in 99.99% of the cases where there is no illness.

After receiving a positive test for the illness the man is devastated, but soon he realises that he is almost certainly not ill.

Why is that if it is 99.99% accurate?

Answer -

1 in a billion people receive a true positive

one in ten thousand people receive a false positive

~99.99% of people receive a true negative

no false negatives, it always detects the illness

The man is in the first two options, it is 100,000 times more likely that it was a false positive. False positives are rare, but the disease is much rarer.