Probability of process success Beforehand my pardon if I have some terminology mistakes since I'm not a mathematician.  
I have a process that consist of multiple activities. Each process $S$ is described by milestones $(k_1, k_2, \ldots, k_n)$ that should be reached at the end of process. Conceptually activity is step of process that reaches  some subset of milestones. Each activity $a$ is represented by triplet: $(K, t, s)$
Where:


*

*K - some subset of $k_i$ milestones - for example $\{k_1, k_3\}$;

*t - total number of activity has been invoked;

*s - succeed number of activity invocation (so $s \leq t$);


The goal to estimate what is the probability of process success if this consists of predefined set of activities $$S = \{a_1, a_2 \ldots a_n\}$$ - it is guaranteed that $S$ fully covers process - so: $$\bigcup_{i}a_i = \{k_1, k_2...k_n\}$$ in other words we are sure that sequence of $a_i$ have no gaps in milestones and all milestones are passed after activities applied. The last note - activities are independent of each other (numbers $t, s$ have no correlation between activities).  
Live example:
Let's assume the Launch cooking is a process. We have exactly 3 milestones for it:


*

*$k_1$ - prepare first dish;

*$k_2$ - prepare second dish;

*$k_3$ - prepare a dessert;


I have following activities:


*

*$a_1$ [go & buy in supermarket] (assume no soup in local one) then $K=\{k_2, k_3\}$. I have already tried this way and very glad with it so: $t=s=100$

*$a_2$ [order in restaurant] there every milestone is available $K={k_1, k_2, k_3}$. But unfortunately I have bad experience with restaurant delivery so $t=10, s=4$

*$a_3$ [cook on my own] I'm not an expert in desert cooking then $K=\{k_1, k_2\}$, also I'm good in cooking but mistakes happen, so $t=10, s=9$


So my question - what is the probability of a successful launch if I combine visiting the supermarket $a_1$ and cooking $a_3$ ? Trivial answer for $a_2$ - success probability is $s/t=4/10=0.4$ - it is clear for me, but how would one combine the probability of multiple activities?
 A: Let's see if this helps:
If each activity has probability of success $p_i = \frac{s_i}{t_i}$ and you can reasonably assume that they are independent then $\mathbb{P}(\text{success } a_1 , a_2) = p_1 + p_2  - p_1 p_2$. This formula extends for $k$ activities by iteration Let ($p_{i_1,\ldots i_k}$ represent $\mathbb{P}(\text{success } a_{i_1}\ldots, a_{i_k})$):
$$\mathbb{P}(a_1, a_2,a_3) = p_{1,2,3} = p_1 + p_{2,3} - p_1 p_{2,3} = p_1 + p_2 + p_3 -p_2 p_3 - p_1(p_2 + p_3 -p_2 p_3) = p_1 +  p_2 + p_3 - p_2 p_3 -p_1p_2 -p_1p_3 + p_1p_2p_3 $$
$$\mathbb{P} (a_1,a_2,\ldots a_k) = \sum_{j \leq k} \sum_{i_1, \ldots i_j} (-1)^{j+1} p_{i_1} \ldots p_{i_j}$$
In the case where these events are not independent you might consider the general approach:
$$ \mathbb{P}(a_1 \cup a_2) = \mathbb{P}(a_1) + \mathbb{P}(a_2) - \mathbb{P}(a_1 \cap a_2)$$
Use the inclusion exclusion principle 
$$\mathbb{P} (\cup_{i=1}^k a_i) = \sum_{j \leq k} (-1)^{j+1} \sum_{i_1 ,\ldots i_j}\mathbb{P}(\cap_{i = 1}^j a_{i_j}) $$
in the case you would like to succeed in the entire process I would consider the probability of achieving each milestone at the time, that is, 
consider the probability $p^j_l = \mathbb{P}(k_j \text{ succeeds  under } a_l)$ use the same arguments (of independence if it is reasonable or inclusion and exclusion principle plus a case by case consideration in case you suspect there are relevant interdependencies between milestones or activities)
