A Game of Thrones fan predicts there is a 70% chance that her favorite character will survive the next season and a 75% chance that her second favorite character will die. There is also a 16% chance that both characters will die. What's the probability that the second character will die given that the first character dies? What kind of probability is this called?

The probability that the second favorite character will die given that the first favorite character dies is 0.53

This kind of probability is called conditional probability

Explanation:

Name the events and their probabiities:

Event A: her favorite character will survive, so P (A) = 0.70

Event B: her her second favorite character will die, so P (B) = 0.75

Both characters will die ⇒ P (B and not A) = 0.16

You want to find P (B | not A).

That is the probability of the succes B (the second favorite character will die) given other event (not A or the first favorite character dies) is certain (it happens) and that is called conditional probability.

P (not A) is the complement probability of A, so P (not A) = 1 - P (A) = 1 - 0.7 = 0.3

So, you have P (B), P (not A) and want to find P (B | not A)

The definition of conditional probability is:

P (X | Y) = P (X and Y) / P (Y)

So, replacing with our terms, we get:

P (B | not A) = P (B and not A) / P (not A) = 0.16 / 0.3 ≈ 0.53