By K.D. Pavate
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Also, N A B is the number of occurrences common to the two events. A Venn diagram is convenient to depict such a situation. We see that the outcomes which constitute event B are made up of two parts. First, there are those outcomes which occurred when A did not occur. Second, there are those which happened when A did occur (NAB). Thus, only the occurrences of A which are simultaneous with those of B are taken into account. Assuming that the sample space is equiprobable, we have: wa P(B/A) - y s both A and B can occur Number of ways A can occur In terms of the frequency definition of probability the unconditional probabilities of events A, B and AB are The conditional probability P(B/A) = NAB NAB/N P(AB) NA NA/N P(A) WITH STRINGS ATTACHED 37 It is of course assumed that P(A) is not zero or else we are landed with an absurd result of P(B/A) becoming infinite.
A word of caution is that not all outcomes are equally likely to happen. MEASURES OF UNCERTAINTY 23 The second definition was based on repeating the random experiment a large number of times and then to estimate the relative frequencies of each outcome. Here the relative frequencies need not be the same for all outcomes. As the number of experiments becomes larger and larger, the relative frequency stabilizes and this is the basis for the second definition of probability. The third definition was based on asking an expert for his considered opinion.
There are 15 different combinations for picking up the four cards. The individual cards A,B,C,D,E and F are present in 10 combinations. The number of ways in which both A and B are present is six. The probability of finding the card B given that A is also present, therefore, happens to be 6/10. It is interesting to realise that when P(B/A) is being computed, we are essentially computing P(B) with respect to the reduced sample space A, rather than with respect to the original sample space S. When calculating the probability of event B, we are really asking ourselves how probable it is that A MATTER OF CHANCE 38 i s WpPj •A A A * A T 0I T Sample space An ace and a king are present together in six ways WITH STRINGS ATTACHED 39 we shall be in event B knowing that we must also be in S.