LOS N requires us to:
calculate and interpret an updated probability using Bayes’ formula
a. Previously, in the ‘total probability rule we discussed, how can we find the probability of a certain event happening, given that some other event has already happened or a given scenario. But at times, the analysts may also require the probability of the scenario, given that the latter event has already occurred. In such situations, we need to work backward to calculate the probability of the former events. The Bayes’ formula allows us to make such calculations.
b. Consider the event tree discussed in the total probability rule:
Here we calculated the total probability of price going up. We summed up the probabilities of both expansion and price going up, and recession and price going up. Thus the total probability was:
Now consider the reverse situation, where we are given the probability of the price going up is 0.62 and are asked to calculate the probability of the economy going through the expansion. This means that we have to find the value for P(S|A); earlier we found the value of P(A|S).
As per the Bayes’ formula, we can find the probability of the proceeding event if we know the probability of the succeeding event and the total probability by using the following equation:
Thus, for the above example, the probability of the economy going through an expansion, given that the price of asset A has already gone up would be:
Similarly, the probability of the economy running into recession, given that the price of asset A has already gone up would be: