Introduction to probability (5) Conditional probability Theorem: If in an experiment the event A and B can be occur then:

Introduction to probability (5) Example1: In a fuse box containing 20 fuses of which 5 are defective. If 2 fuses are selected at random and removed from the box without replacing the first, what is the probability that both fuses are defective?

Introduction to probability (5) Solution: Let A be the event that the first fuse is defective. Let B as the event that the A occurs and then the B occurs after A has occurred

Introduction to probability (5) Definition: Two events A and B are independent if and only if Assuming that the existence of conditional probabilities. Otherwise A and B are dependent.

Introduction to probability (5) Example2: Consider an experiment in which 2 cards are drawn in play card with replacement in succession for the events are: A: the first card is ace. B: the second card is spade بستوني

Introduction to probability (5) Solution: Then the event A and B are independent

Introduction to probability (5) Example3: One bag contains 4 white balls and 3 black balls, a second bag contains 3 white ball and 5 black balls. One ball is drawn from the first bag and placed unseen in the second bag. What is the probability that a ball now drawn from a second bag is black?

Introduction to probability (5) Solution:

Introduction to probability (5) Let B1, B2, W1 represent respectively the drawing of a black ball from bag1, a black ball from bag2 and a white ball from bag1. We are interesting in the union of the mutually exclusive events

Introduction to probability (5) Theorem: Two events A and B are independent if and only if Therefore to obtain the probability that two independent event will both occur we simply find the product of their individual probabilities.

Introduction to probability (5) Example4: 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: The entire system works and The component C does not work given that the entire system works. Assume that the four components work independently.

Introduction to probability (5) Solution:

Introduction to probability (5) From diagram the probability that the entire system works can be calculated as:

Introduction to probability (5) 2.