The current share price quoted to 30 €. The volatility is 25% per annum. The drift of 5% per annum
1) How is the share price in 6 months probabilistic distributed?
2) The expected value of the stock price in 6 months giving.
Regarding 1) I used this (http://en.wikipedia.org/wiki/Geometric_Brownian_motion#Solving_the_SDE) getting a result of 0,1862.
For 2) I used (30⋅e^0,05⋅0,25) getting a result of 33,9944.
I'm not sure whether or not I used the correct formula and hope one could help/explain.
How can your answer to 1 be a number? Isn't it asking for a distribution? Geometric brownian motion results in a time-dependent lognormal distribution of share prices, as stated in your wikipedia link in the properties section. Here, $\sigma = 0.25\rightarrow {\sigma}^2=.0625,\mu=.05$
You need to plug in the above values into the lognormal function shown in the above link.
For 2) the answer is also found in the link I gave you, see the "expected value" section, which gives $E(S_t)=S_0e^{\mu t}$, which as you will see does not depend on the volatility. The result for a 6 month horizon is $S_{0.5}=30e^{.05*.5}= 30.79$ which makes sense given the rather low return rate for this security -- we would expect to have approx. 31.5 at the end of the year. Note that your answer of 33 (which is not the result of your calculation) implies an expected return of 10% in 6 months, or approximately 20% per year, which is 4x the stated growth rate.