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

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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?

3. 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

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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.

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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

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Solution:
Then the event A and B are independent

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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?

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Solution:

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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

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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.

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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.

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Solution:

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

14. Introduction to probability (5)2.