I was given a conditional probability task with the following wording:
The probability of having black hair is 60%. The probability of having blue eyes if your hair is not black is 40%. The probability of having blue eyes given that your hair is black is 10%. What is the probability of having black hair if your eyes are blue?
I am not a native speaker so I was conflicted if the task implies conditional probability or joint probability. Anyway I formulated the task as follows:
P(BH) = .6
P(BE | not BH) = .4
P(BE | BH) = 0.1
P(BH | BE) = ?
1. Is this correct? Did I understand the wording correctly?
After formulating these equations I could derive the following:
P(BH) = .6
P(not BH) = .4
==================
P(BE | not BH) = .4
P(BE and not BH) / P(not BH) = .4
P(BE and not BH) / .4 = .4
P(BE and not BH) = 0.16
==================
P(BE | BH) = 0.1
P(BE and BH) / P(BH) = 0.1
P(BE and BH) / 0.6 = 0.1
P(BE and BH) = 0.06
==================
P(BH | BE) = P(BH and BE) / P(BE)
2. Now I am stuck as I don't know how I could get the P(BE) value. How to get value of P(BE)?
This question is most readily answered by the use of Bayes theorem. From Bayes we have $$ \mathbb P(BH\mid BE) = \frac{\mathbb P(BE\mid BH)\mathbb P(BH)}{\mathbb P(BE)}. $$ We are given all of these quantities aside from $\mathbb P(BE)$, but we can compute $\mathbb P(BE)$ using the law of total probability and definition of conditional probability: \begin{align} \mathbb P(BE) &= \mathbb P(BE\cap BH)+\mathbb P(BE\cap BH^c)\\ &=\mathbb P(BE\mid BH)\mathbb P(BH) + \mathbb P(BE\mid BH^c)\mathbb P(BH^c)\\ &= \frac1{10} \cdot\frac35 + \frac25\cdot \frac25\\ &= \frac{11}{50} \end{align} It follows that $$ \mathbb P(BH\mid BE) = \frac1{10}\cdot \frac35 \cdot\frac{50}{11} = \frac3{11}. $$