Answer: yes, the first, because it overlaps itself. You can have two occurrences of it in 5 tosses.

Everybody gets this wrong.

Same thing with DNA, and cutting chunks with restrictions enzymes.

99% right HIV test.

A person is picked at random and tested for the disease.

The test gives a positive result.
What is the probability that the person actually has the disease?

**Not** 99%.
It depends on how common or rare the disease is.

Suppose the disease affects 1 person in 10,000.

Out of one million people, 100 have the disease, and 999,900 haven't.

Testing the 100 yields 99 positive.

Testing the 999,900 yields 9,999 positive.

So, the correct answer is less than 1%.

We are not good at reasoning with uncertainty.

If there's something we're bad at doing,
at the very least we should recognize it, and we tend not to.

Conferences Marc Girod