probability
Three radar sets, operating independently, are set to detect any aircraft flying through a certain area. Each set has a probability of 0.02 of ailing to detect a plane in its area. If an aircraft enters the area, what is the probability that it:
a.) goes undetected?
P(undetected | Area) = (0.02)^3
b.) is detected by all three radars?
P(Detected | Area) = 1- (0.02)^3
Is this correct? I am unsure about this.
You are correct (again).
Here is the solution I found, maybe wrong:
All three components working happens with probability $p^3$.
Exactly two components working happens with probabiliy $3p^2(1-p)$.
So at least two components working happens with probability $p^3+3p^2(1-p)=3p^2-2p^3$.
Now, since $p$ itself is randomly distributed, our answer is just $\displaystyle \int_{0}^{1}(3p^2-2p^3)\,dp=0.5$.
In fact, this is kind of intuitive. There is the same chance that $2$ or $3$ components fail, as there are $2$ or $3$ components work.
Best Answer
As André pointed out B is incorrect.
You already said that the probability none of them works is $0.02^3$ so if the probability all three works is $1-0.002^3$ then the probability anything else happens is 0. So either all work or none works (this already feels wrong).
To see the probability none work you need to take $(1-0.002)^3=0.98^3$ since it is the probability each one doesn't work.