In a book by Wayne Winston for operations research I found this question.
Here's how I did it:
Let $t$ be the no.of subjects to pass and let h be the no.of hours she has in hand for studying.
Then $f_t(h)$ be the probability of passing subjects t,t+1,..3 when h hours are in hand. Let $x_t$ be the no.of hours spent on subject t.
$0<=x_t<=h; h\in {0,1,2,3,4}$
Then $f_3(4)=0.5 \\ f_3(3)=0.44 \\f_3(2)=0.4\\f_3(1)=0.3\\f_3(0)=0.1$
Let $P_t(x_t) $ be the probability of passing subject t when $x_t$ hours are spent on it.
$f_t(h)=max ${${P_t(x_t)}+f_{t+1}(h-x_t)$}
According to this I get
$f_(4)=$max$P_1(0)+f_2(4)=0.98\\P_1(1)+f_2(3)=1\\P_1(2)+f_2(2)=1\\P_1(3)+f_2(1)=0.93\\P_1(0)+f_2(4)=0.425$
The answer I get is she should study French for 1 hour, German for 1 hour and Stat for two hours. But the answer given in the book says French 1 hour, German 0 hours, Stat 3 hours.
What's wrong with my method? And what is the probability of passing at least one subject ?
I have a different approach.
indices:
$ij$: The amount of hours i Angie is learning for course j.
$i \in \{0,1,2,3,4\}$
$j \in \{1,2,3\}$
variables:
$$x_{ij}=\begin{cases} 1, \ \texttt{if Angie learns i hours for course j} \\ 0, \ \texttt{else}\end{cases}$$
$$f_{ij}=\texttt{probability, that Angie passes course j, if she is learning i hours}$$
constraint 1: Angie is learning 4 hours a week
$$\sum_{i=0}^4 \sum_{j=1}^3 i \cdot x_{ij}=4$$
constraint 2: Only one hours/course combination for each course
$$ \sum_{i=0}^4 x_{ij}=1 \ \ \ \ \forall j \in \{1,2,3\}$$
objective function:
Because of the comments of joriki the objective function is
$$\texttt{max} \ \ \ 1-\prod_{j=1}^3 \left(1- \sum_{i=0}^4 f_{ij} \cdot x_{ij} \right) $$