7-5 - Chapter 7.5 Conditional Probability • In some cases...

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Unformatted text preview: Chapter 7.5: Conditional Probability • In some cases, we may perform an experiment, and already know something about the outcome. • For example, suppose I had a bag containing 5 red marbles and 5 green marbles. My experiment will consist of selecting two marbles from the bag, one at a time, without replacement. • Suppose I know that the first marble is red. Then the probability that the second marble will also be red will be 4 9 , as there will be four remaining red marbles out of nine remaining marbles. • On the other hand, if the first marble is green, then the probability that the second marble will be red will be 5 9 , as there will be five red marbles out of the nine remaining marbles. • These probabilities are known as conditional probabilities . They are the probability of an event occurring, taking into account some information about the experiment. • The information which is known is sometimes called the condition . • Suppose an experiment has two events A and B . Suppose we know that event A has occurred. We write the probability that B will occur, taking the occurrence of A into account, as P ( B | A ). • We read P ( B | A ) as “the probability that B will occur given that A occurs”. • Normally, when we find P ( E ), we find two things: the number of outcomes in E , and the total number of possible outcomes (the sample space). • The outcomes in B | A are precisely those in which B occurs, and A also occurs. In other words, the number of outcomes in B | A is n ( B ∩ A ). • If we know that A occurs, then the total number of possible outcomes will be the number of outcomes in A , which is n ( A ). • This gives us: P ( B | A ) = n ( B ∩ A ) n ( A ) • Now, suppose that I were to divide the top and bottom of this fraction by n ( S ). This would give me: P ( B | A ) = n ( B ∩ A ) n ( S ) n ( A ) n ( S ) 1 • This gives us the formula for conditional probability: P ( B | A ) = P ( B ∩ A ) P ( A ) . Note that it is the condition (what we know has occurred) which is in the denominator. If A and B are events, and P ( A ) negationslash = 0, then the probability that B will occur if it is known that A has occurred is written as P ( B | A ), and P ( B | A ) = P ( B ∩ A ) P ( A ) Example: Suppose I roll two dice, and that the first roll was a five....
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This note was uploaded on 02/05/2011 for the course MATH 377 taught by Professor Stephenlang during the Spring '11 term at University of Victoria.

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7-5 - Chapter 7.5 Conditional Probability • In some cases...

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