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Example1:
An electrical system consists of four components. The system works if component A and B work and either of the component C or D works. The probability of A working is 0.9 and B is 0.9, C is 0.8, D is 0.8 find the probability that:
1. The entire system works and
2. The component C does not work given that the entire system works. Assume that the four components work independently.

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Solution:
From diagram the probability that the entire system works can be calculated as: 2.

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Example2:
Suppose the diagram of an electrical system is as given in diagram below, what is the probability that the system works? Assume that the components fail independently.

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Solution:
P = (0.95)(1-(1-0.7)(1-0.8))(0.9)=0.8037

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Example3:
A small town has one fire engine and one ambulance a available. The probability that ambulance is available when call is 0.92 and the probability of fire engine available is 0.98.
In the event of an injury resulting from a burning find the probability that both ambulance and fire engine will be available. Assuming they operate independently.

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Solution:
Let A and B represents that the fire engine and ambulance are available

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Example4:
In an experiment to study the relationship of hypertension and smoking habits, the following data are collected for 180 individuals, where H and NH stands for Hypertension and No hypertension respectively . if one of these individuals is selected at random find that the person is:

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Experiencing hypertension, given that the person is heavy smokers.
A non smoker, given that the person is experiencing no hypertension
Smoking
Non
Moderate
Heavy
Total H 21 36 30 87 NH 48 26 19 93 69 62 49
180

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Example5:
A manufacturer of a flu vaccine concerned about the quality of its flu serum. Batches of serum are processed by three different departments having rejection rates 0.1, 0.08 and 0.12 respectively. The inspections by the three departments are sequential and independent.

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What is the probability that the batch of serum survives the first department inspection but is rejected by the second one?
What is the probability that the batch of serum is rejected by the third departments?
Solution:

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Example6:
The probability that an automobile being filled with gasoline also needs an oil change is 0.25; the probability that it needs a new oil filter is 0.40; and the probability that both oil and filter need changing is 0.14.
a) If the oil has to be changed, what is the probability that a new oil filter is needed?
b) If a new oil filter is needed, what is the probability that the oil has to be changed?

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Solution:
Let that
O : oil change is needed.
F: an oil filter is needed.