1 Conditional Probability STA 281 Fall 2011 1 Definition Often we are only interested in particular rows or columns of a probability table. Consider the newspaper example, and the question “Of those that receive the morning paper, what proportion receive the evening paper?” This question does not concern the entire population of households; it only concerns those who receive a morning paper. Probabilities that refer only to subsets of the population are called conditional probabilities . Recall the probability table we constructed 0.10 0.20 0.30 0.50 0.20 0.70 0.60 0.40 1.00 The question asked concerns only those who receive a morning paper, which is 60% of the entire population. We want to know, out of that 60%, what proportion receive the evening paper. To provide an intuitive fell for how this question is answered, suppose we sampled 100 people from the population. On average 60 of those people would receive the morning paper. Looking at the M column of the table, we see that of those 60, on average 50 receive only the morning paper while 10 receive both. So 10 of the 60 who receive the morning paper also receive the evening paper, so the conditional probability is 10/60=1/6. Usually we don’t go through the argument concerning sampling a set of people and just divide the probabilities directly. There are 60% of the people who receive a morning paper, with 50% receiving only the morning paper and 10% receiving the evening paper. So 0.10/0.60=1/6 is the conditional probability of receiving an evening paper given one receives a morning paper. Mathematically, a conditional probability has two parts: First, a conditional probability only asks about a subset of the population, not the entire population. Second, a conditional probability asks some property of that subset. In our example question, the subset of interest was those who receive the morning paper, while the property we are interested in was receiving an evening paper. In general, we have a question: “of those who are in subset A, what is the probability they are in B.” This question is translated into mathematical symbols , which is read “the probability of B given A.” Notice how we solved the problem. First, we found which of the individuals were in the subset of interest. This involved finding , the unconditional probability of the subset. Then, within that subset , we found how many individuals had the property we were interested in. The result was Instead of writing this fraction in wo rds, we can use symbols. For the denominator, the “people in subset A” refers to . For the numerator, the people must be in subset A, but they must also have property B. Since both criteria must be satisfied, the numerator is , resulting in Mathematically, this formula is the definition of conditional probability. Rearranging the terms immediately implies what is called the intersection rule
2 Similarly, since just switching the roles of A and B in the definition of conditional probability yields , we find Since is the same as , the two previous equations provide two ways of finding the probability of an intersection.
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