Introduction to probability (6) 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.

Introduction to probability (6)

Introduction to probability (6) Solution: From diagram the probability that the entire system works can be calculated as: 2.

Introduction to probability (6) 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.

Introduction to probability (6) Solution: P = (0.95)(1-(1-0.7)(1-0.8))(0.9)=0.8037

Introduction to probability (6) 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.

Introduction to probability (6) Solution: Let A and B represents that the fire engine and ambulance are available

Introduction to probability (6) 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:

Introduction to probability (6) 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

Introduction to probability (6) 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.

Introduction to probability (6) 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:

Introduction to probability (6) 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?

Introduction to probability (6) Solution: Let that O : oil change is needed. F: an oil filter is needed.