1000 people are grouped by their age and hobby. The table is the following:
\begin{array} {|r|r|}\hline & 18-22 & 23-27 & 28-32 & 33-37 & total \\ \hline Music & 54 & 40 & 42 & 66 & 202 \\ \hline Movies & 77 & 68 & 90 & 70 & 305 \\ \hline Walking & 28 & 43 & 50 & 52 & 173 \\ \hline Other & 90 & 78 & 71 & 81 & 320 \\ \hline Total & 249 & 229 & 253 & 269 & 1000 \\ \hline \end{array}
You meet a 19-year-old girl who took the survey. What is the probability she prefers music?
My approach:
You meet the girl and that girl has $54/249$ of preferring music, that's it.
The question is: Should you use Bayes?
$P(Music|19y) = \frac{P(Music)*P(19y|Music)}{P(19y)}$
$P(Music) = \frac{202}{1000}$
$P(19y|Music) = \frac{54}{249}$
$P(19y) = \frac{249}{1000}$
This would lead to $P = \frac{1212}{6889}$
You mixed up the denominator for $P(18-22|Music)$, which should equal $\frac{54}{202}$. Then Bayes' Theorem ALSO gives you the right answer: $P(music|18-22) = \frac{P(Music)P(18-22|music)}{P(18-22)}=\frac{\frac{54}{202}\frac{202}{1000}}{\frac{249}{1000}} = \frac{54}{249}$. That said, the first way of doing it is faster.