Acertain system can experience three different types of defects. let ai (i = 1,2,3) denote the event that the system has a defect of type i. suppose that the following probabilities are true. p(a1) = 0.14 p(a2) = 0.09 p(a3) = 0.09 p(a1 ∪ a2) = 0.16 p(a1 ∪ a3) = 0.18 p(a2 ∪ a3) = 0.14 p(a1 ∩ a2 ∩ a3) = 0.02 (a) what is the probability that the system does not have a type 1 defect? (b) what is the probability that the system has both type 1 and type 2 defects? (c) what is the probability that the system has both type 1 and type 2 defects but not a type 3 defect? (d) what is the probability that the system has at most two of these defects?

a.

b.

c. By the law of total probability,

According to the inclusion/exclusion principle,

but by DeMorgan's law,

So

and from there we find

d. The event of having at most two of these defects is complementary to the event that all three defects occur simultaneously:

the answer is:

the answer is:

to solve the problem, we need to remember the quotient of power property, it's defined by the following relation:

if we have a quotienf of powers that have the same base, we need to keep the same base and subtract the exponent of the denominator power to the exponent of the numerator power.

so, we are given the expression:

then, calculating we have:

hence, the answer is:

have a nice day!

c

step-by-step explanation: