If you are calculating the probability of being able to prevent the occurrence of an event and have already determined the following:

. Barrier 1 has a 50% chance of preventing the problem
. Barrier 2 has a 50% chance of preventing the problem
. Barrier 3 has a 75% chance of preventing the problem

What is the combined effect of all three barriers, with regard to probability of preventing the problem? I don't need the answer, I need the formula.


Looking at a "similarly different" problem, from the probability of occurrence of an event if:

. Barrier 1 reduces the likelihood of occurrence to 1 in 100 (0.01)
. Barrier 2 reduces the likelihood of occurrence to 1 in 1000 (0.001)
. Barrier 3 reduces the likelihood of occurrence to 1 in 10000 (0.0001)

What is the combined effect of all three barriers, with regard to probability of occurrence of the event? I don't need the answer, I need the formula.

Thank you.

If you can assume that the barriers act independently, and that if it's true that it only requires one barrier to be successful at stopping the problem, then you can think through it like this:

Let P_n = the probability that barrier n stops the problem, and let \bar P_n = the probability that barrier n fails to stop it. The probability that at least one of the three barriers working in series stopping the problem is 1 minus the probability of none of the three barriers together stopping the problem. If the barriers work independently, the probability of all three barriers failing to stop it is

\bar P{all} = \bar P_1 \times \bar P_2 \times \bar P_3.

So you get:


P(\text{at least one barrier working}) = 1 - \bar P_{all} = 1- (1-P_1)(1-P_2)(1-P_3)

Can you take it from here?

I have it. Thanks.