Precal Probability Homework > The letters in the word AARDVARK are printed on square pieces of cardboard with one letter per card. > > The eight letters are placed in a hat and one letter is chosen at random. Find the following probabilities: > > a) P(the letter chosen is a vowel given that the letter falls in the first half of the alphabet) So this is a homework problem I'm having trouble with... the teacher has provided an answer (different from mine!) but not an explanation, and he'll be gone a while, so I'd like to know what I did wrong. The probability that the letter falls in the first half of the alphabet is $5/8$, as there are 3 As, 1 D and 1 K. The probability that the letter chosen is a vowel from AARDVARK is $3/8$. Then using conditional probability, I get $(5/8)*(3/8) / (5/8) = (3/8)$. However, the answer is apparently $3/5$. What did I do wrong?

The conditional probability $P(A|B)$ is given by the formula $\frac{P(A\cap B)}{P(B)}$. That numerator is the probability of choosing a vowel that is also in the first half of the alphabet $\left(\frac38\right)$. The denominator is simply the probability of choosing a letter in the first half of the alphabet $\left(\frac58\right)$.

It looks like you tried to use the formula $P(A\cap B)=P(A)P(B)$. That only works for independent events, and in this case, choosing a vowel and choosing a letter in the first half of the alphabet are _not_ independent.