Conditional Probability Sundry Computational Tools Lecture 6/10
Chance has its reasons. Conditional Probability When the choice of a sample space is not self evident (that is, the set of all possible outcomes being considered), it is worthwhile to use the symbol P(A|B) read "the probability of A given B", to denote the conditional probability of the event A with respect to the sample space B. If B is an event from a sample space S, then shrinking the sample space S to a new set of outcomes B refers to the probability that A will occur given that B has already occurred. A Discrete Example from our text. Suppose that all of the freshmen of an engineering college took calculus and discrete math last semester. If 70% of the students passed calculus, 55% passed discrete math and 45% passed both, what is the probability that a randomly selected freshman that passed calculus also passed discrete math? A Continuous Example Suppose that a bus arrives at a station everyday between 1:00 P.M. and 1:30 P.M. If the bus has not arrived by 1:10, what is the probability that it will arrive in the next ten minutes? Definition If P(B) > 0, the conditional probability of A given B, denoted by P(A|B), is  
 Do you recall the Monty Hall Game from our first week? Let's answer the question about whether it is better to stick or switch. Law of Multiplication The defining equation for conditional probability may also be written as: This formula is useful when the information given to us in a problem is P(B) and P(A|B) and we are asked to find P(AB). An example illustrates the use of this formula. Suppose that 5 good fuses and two defective ones have been mixed up. To find the defective fuses, we test them one-by-one, at random and without replacement. What is the probability that we are lucky and find both of the defective fuses in the first two tests? Theorem If P(A1A2A3...An-1) > 0, then Law of Total Probability Use the following diagram to calculate P(A) in terms of P(B), P(A|B), P(A|Bc) and P(Bc).
 The result P(A) = P(A|B)P(B) + P(A|Bc)P(Bc), is called the Law of Total Probability. As we have seen, the various representations of probabilities in terms of sets are useful for calculation when the information given to us in a particular problem matches the set structure of a formula. For an example of a problem that is solved using the law of total probability, suppose an insurance company rents 35% of the cars for its customers from agency I and 65% from agency II. If 8% of the cars from agency I and 5% of the cars from agency II breakdown during the rental periods, what is the probability that a car rented from this insurance company breaks down? We can generalize the law of total probability by defining the partition of a set. Definition Let A1,A2, A3, ... , An be a set of nonempty subsets of a sample space S. If the events A1,A2, A3, ... , An are mutually exclusive and È Ai = S, then the sets A1,A2, A3, ... , An are called a partition of S.
Theorem If A1, A2, A3, ... , An is a partition of the sample space of an experiment and P(Ai ) > 0 for i = 1, 2, 3, ... , n, then Illustrate the theorem. In practice, if the number of partitions is not too large, a tree diagram often works well for solving the problem. Solve the following problem with the use of a tree diagram: An urn contains 10 white and 12 red chips. Two chips are drawn at random and, without looking at their colors discarded. What is the probability that the third chip drawn is red? T he Gambler's Ruin Problem is one of the classic problems of probability.
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A Partition of S
