A group is made up of ten people: four from Modena and six from Parma. A person from the group, chosen at random, writes the name of their city. A letter of the word written in this way is chosen at random which turns out to be a vowel. What is the probability that the person who wrote is from Parma?
I think I had to calculate $ P(P|V) = \frac{ P(V|P)P(P) }{P(V) } $ where $ P(P) $ is the probability that a person is from Parma, so $ \frac{6}{10} $. $ P(P) $ is the probability that a letter is a consonants or a vowel so $ \frac{1}{2} $ and $ P(P|V) = \frac{3}{6} $ because this is the probability to have a vowel if we choosed a person from Modena. So I obtained that $ P(P|V) = 0.72 $ but I don’t know if is correct
If a person is from PARMA the probability to get a vowel is $\frac{2}{5}$, not $\frac{1}{2}$
Asssuming that a person doesn't lie when writing down the name of his City, the requested probability is
$$\mathbb{P}[\text{PARMA}|V]=\frac{\frac{6}{10}\cdot\frac{2}{5}}{\frac{6}{10}\cdot\frac{2}{5}+\frac{4}{10}\cdot\frac{1}{2}}=\frac{6}{11}\approx0.545$$