Intuitively we might think that a new user in the 30-35 age group's probability to convert [P(conversion | in the 30-35 age group)] is the same as the conversion rate [P(in the 30-35 age group | conversion)]
err... your intuition is wrong? $P(A \cap B) \neq P(A|B) \neq P(B|A)$. Let's work on your intuition here before moving on. An equivalent analogy would be like saying:
\begin{equation}
\begin{split}
X &= P(\text{anarchist}| \text{in the 18-25 age group})\\
Y &= P(\text{in the 18-25 age group}|\text{anarchist})\\
X &= Y
\end{split}
\end{equation}
$X$ is essentially the proportion of people of a certain age who are anarchists. $Y$ is essentially the proportion of anarchists who are in a certain age group. Focusing our attention to the US, we can safely say that most people of voting age are democrat or republican, so $X$ will be small. But, most people who are anarchists are idealistic young people (to generalize broadly), so $Y$ will actually be large. Clearly, $X \neq Y$. I believe your confusion may lie in equating the notation used for $X$ and $Y$ with the following quantity:
$$Z = P(\text{anarchist} \bigcap \text{in the 18-25 age group})$$
Here, $Z$ is the proportion of people who are both anarchists and in our age group over the entire population. When we take a conditional probability, we are essentially constraining our sample space. $Z$ is not conditioned on anything, it's just an intersection, so we have to consider the entire sample space. Although $Z$ is different from $X$ and $Y$, it is closely related to those quantities by the law of conditional probability:
$$P(A | B) = \frac{P(A \cap B)}{P(B)}$$
Therfore,
\begin{equation}
\begin{split}
X&=\frac{Z}{P(\text{in the 18-25 age group})} \\
Y&=\frac{Z}{P(\text{anarchist})} \\
\end{split}
\end{equation}
So as small as $X$ may be, we know that $Z$ is necessarily smaller (unless one of your conditioning probabilities equals 1).
I hope this clarifies your confusion a bit.
Is this the correct way to calculate the probability to convert for a new user in the 30 - 35 age group?
Well, your application of Bayes theorem looks fine, but you could make your life a bit easier since you own the data. Your target is just a conditional probability, so let's focus on that:
$P(\text{conversion} | \text{in the 30-35 age group}) = $
$=\frac{P( \text{conversion} \bigcap \text{in the 30-35 age group})}{P(\text{in the 30-35 age group})}$
$= \frac{(\text{number of people who converted AND age 30-35})/(\text{total sample size})}{(\text{number of people age 30-35})/(\text{total sample size})}$
$= \frac{\text{number of people who converted AND age 30-35}} {\text{number of people age 30-35}}$
You can use Bayes theorem as you suggested, but I suspect this latter formulation will be simpler to calculate. You already used an analogous calculation to figure your conversion rate.
