I have a probability theory task, but I cannot get an idea of how one can solve it. Even, I don't understand question well. Any help is appreciated:
Usage of growth hormones $s_1, s_2, s_3$ leads to certain biological effect with probabilities $p_1, p_2, p_3$. There were $n$ experiments using only one of the hormones. The probabilities that in the experiments was used $s_1, s_2, s_3$ are $W_1, W_2, W_3$ respectively. The positive effect was in $m$ experiments. What is the probability that $s_1$ was used in the experiments?
Given:
\begin{align} P(\textrm{Positive effect}|s_i)&=p_i,\quad i\in\{1,2,3\}\\ P(s_i)&=W_i,\quad i\in\{1,2,3\} \end{align}
Assuming independent trials: \begin{align} P(\textrm{Positive effect in $m$ out of $n$ experiments}|s_i)&=C_m^np_i^m(1-p_i)^{n-m},\quad i\in\{1,2,3\} \end{align}
Using Bayes theorem: \begin{align} P(s_1|\textrm{Positive effect in $m$ out of $n$ experiments})=\frac{P(\textrm{Positive effect in $m$ out of $n$ experiments}|s_1)P(s_1)}{\sum_{i=1}^3 P(\textrm{Positive effect in $m$ out of $n$ experiments}|s_i)P(s_i)} \end{align}