A certain 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) = 0.10
P(A3) = 0.07
P(A1 ∪ A2) = 0.16
P(A1 ∪ A3) = 0.17
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?

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khansawaqar7 khansawaqar7 · Answer 1 · 2020-02-04T10:52:16+01:00

Answer:

a. 0.86

b. 0.08

c. 0.06

d. 0.98

Step-by-step explanation:

a.

The probability that system doesn't have type 1 defect is

P(A1')=1-P(A1)=1-0.14=0.86

Thus, the probability that system doesn't have type 1 defect is 86%.

b.

The probability that system have both type 1 defect and type 2 defect

P(A1∩A2)=P(A1)+P(A1)-P(A1∪A2)

P(A1∩A2)=0.14+0.1-0.16=0.08

Thus, the probability that system have both type 1 defect and type 2 defect is 8%.

c.

The probability that system have both type 1 defect and type 2 defect but not type 3 defect

P(A1∩A2∩A3') =P(A1∩A2 )-P(A1∩A2∩A3)

P(A1∩A2∩A3') = 0.08-0.02=0.06

Thus, the probability that system have both type 1 defect and type 2 defect but not type 3 defect is 6%.

d.

The probability that system has at most 2 defectives

P(At most 2 defectives)= 1-P(3 defectives)

P(3 defectives)=P(A1∩A2∩A3)=0.02

P(At most 2 defectives)= 1-P(3 defectives)=1-0.02=0.98

The probability that system has at most 2 defectives is 98%.