# Probability for at least 2 of 3 events happening whereas 2 are dependent and 1 is independent

We have 3 pumps, 2 electric pumps and one turbin pump. What is the probability of at least 2 of these pumps failing at the same time?

$E_j = Electric\ pump\ j\ is\ faulty\ condition,\ j=1,2$

$P(E_1)=P(E_2)=0.01,\ P(E_2|E_1)=0.1$

$T = The\ Turbin\ pump\ is\ in\ faulty\ condition$

$P(T)=0.04$, T is independent of the electric pumps

The answer is 0.00172 according to the solution set given at the end of the task, but I can't seem to find the correct calculation to get the answer.

• What calculations have you tried, then? – Arthur Sep 12 '18 at 11:59

## 1 Answer

The classical inclusion-exclusion principle should give us the answer.

2 electric ones failing at the same time: $0.01\times0.1=0.001$

one electric and one Turbin failing: $2\times0.01\times0.04=0.0008$

all three failing: $0.001\times0.04=0.00004$

Hence the answer is $0.001+0.0008-2\times0.00004=0.00172$ since we considered the case all three failing three times.