Conditional Probability
Suppose a dice was rolled, and we want to know the probability that the outcome was 4. Assume that the six outcomes are equiprobable. Clearly, the answer is one-sixth. Now, suppose a dice was rolled, and we were told that the number was even. Again, we want to know the probability that the outcome is 4. We realize that since only 2, 4, or 6 are the possible outcomes, the probability that 4 has appeared must be larger than one-sixth. As a matter of fact, the reader has probably arrived at the answer: the probability that 4 has appeared is one-third.
Consider again the problem of bald people. Suppose we choose a person at random. As shown above, the probability that the person is bald is 0.392. Suppose we were told that this person was a female. Then we would say, at least intuitively, that the probability of this person being bald would be less than 0.392. On the other hand, if we were told that the person was male, then the probability that he is bald would be greater than 0.392.
These two examples bring up the notion of conditional probability of an event. Let S be a sample space and A and B be two events in S. The probability that event A occurs given that event B has occurred is defined as the conditional probability of event A given the occurrence of event B, which is denoted p(A|B). In the example of rolling a dice, let A denote the event, “the outcome is 4”, and B denote the event, “the outcome is an even number.” The conditional probability p(A|B) is then equal to 1/3. In the example of bald people, let A denote the event that a bald person is chosen, B denote the event that a female is chosen, and C denote the event that a male was chosen, the chance is better that a bald person was chosen is greater than the ability p(B|A) of a woman being chosen given that a bald person was chosen is less than the conditional probability p(C|A) of a man being chosen given that a bald person was chosen. We shall show how to compute these conditional probabilities to confirm all of these intuitive notions later.
The occurrence of event B has effectively changed the probabilities associated with the samples in the sample space. Obviously, the probability associated with a sample not included in event B increases. Let us examine again the associated with a sample included in event B increases. Let us examine again the simple example of rolling a dice. If we were told that an even number appeared, the probabilities of the samples 1, 3, and 5 all become 0, since it is certain that none of them could have occurred. On the other hand, the probabilities of the samples 2, 4 and 6 become one-third. Thus, indeed,
p(4 appeared |even number appeared) = 1/3
In general, let p_{B}(x_{i}) denote the probability associated with sample x_{i} given that event B has occurred. As pointed out above, for x_{i} ∉ B, p_{B} (x_{i}) = 0. However, for the samples in event B their relative frequencies of occurrence remain the same while the sum of their probabilities should equal to 1, that is, Σx_{i} ϵ B p_{B}(x_{i}) = 1.
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