False proofs that look correct

I remember seeing a list of False Proofs when I was taking Discrete Maths and I found it to be very interesting and also helpful.

So, if anyone knows some common proof mistakes students make or some cool sneaky way to trick people in a proof send them my way! Bonus points if they're cs/algorithms related.

Thank you so much!

One of my favourites is the "brothers paradox":
https://en.wikipedia.org/wiki/Boy_or_Girl_paradox

I tell it as I learned it*, as follows:
in a village, each family has two children, elder and younger. The children are born with probability $\frac12$ for either boy or girl.

I knock on a door, and a boy opens it and says "I'm the elder". What is the probability he has a sister? (clearly $\frac12$).

I knock on another door, and a boy opens it and says "I'm the younger". What is the probability he has a sister? (clearly $\frac12$).

I knock on a third door, and a boy opens it and says nothing. What is the probability he has a sister?

Now the lies start:
if I use the law of total probability, I might say something like:
$$Pr[sister]=Pr[sister|elder]\cdot Pr[elder]+Pr[sister|younger]\cdot Pr[younger]=\frac{1}{2}Pr[elder]+\frac{1}{2}Pr[younger]=\frac12$$
But on the other hand, I could say that the only case where he does not have a sister is the (boy,boy) setting, and we already know this house is not (girl,girl), so the probability is $\frac23$.

Both are very convincing, if told correctly.

The fallacy is that the sample space is undefined, or put more whimsically "How does the family decide who answers the door?"

*I learned it this way from the legendary Prof. Raz Kupferman.

StackExchange Computer Science Q#160559, answer score: 18