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

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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
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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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