This scheme of water flow contains 7 elements, each of element works with let's say 0,8 probability, they work independently. How to find probability that scheme works? Without 1st element and 7th element scheme don't work, it should start with 1st, then flow can go to 2,3 or 4 or 5, 6 and then reach 7, so I tried to calculate it like this:
$$e_1(e_2e_3+e_4+e_5e_6 - e_2e_3e_4e_5e_6)e_7$$ but my result is over 1, so my calculation isn't proper. I totally know how it should be solved without that 4th element in the middle, it would look like:
$$e_1(e_2e_3+e_5e_6 - e_2e_3e_5e_6)e_7$$
So I want to ask what I am doing wrong with that 4th element and how to find correct probability?
Let e.g. $E_{1,3,4}$ denote the event that $e_1,e_3,e_4$ work.
Then to be found is the probability :$$P(E_1\cap(E_{2,3}\cup E_4\cup E_{5,6})\cap E_7)=P(E_1)P(E_{2,3}\cup E_4\cup E_{5,6})P(E_7)$$
We can find $P(E_{2,3}\cup E_4\cup E_{5,6})$ by means of the principle of inclusion/exclusion:
$$P(E_{2,3}\cup E_4\cup E_{5,6})=$$$$P(E_{2,3})+P(E_4)+P(E_{5,6})-P(E_{2,3,4})-P(E_{2,3,5,6})-P(E_{4,5,6})+P(E_{2,3,4,5,6})$$
Try to finish this yourself.