LOS E, F, and G requires us to:
e. explain the multiplication, addition, and total probability rules;
f. calculate and interpret 1) the joint probability of two events, 2) the probability that at least one of two events will occur, given the probability of each and the joint probability of the two events, and 3) a joint probability of any number of independent events;
g. distinguish between dependent and independent events
a. Consider a scenario, where we may expect an economy may either run into recessions or go through an expansion phase, and both the possibilities have a probability of 30% and 70% respectively. In each of the above cases, the price of the share may either go up or down. The probabilities in each case would be as follows:
Now, if we have to calculate the total probability of price going up, we would have to sum up the probabilities of both expansion and price going up, and recession and price going up. Thus:
We know that
and
Therefore, the total probability of an increase in the price of the stock is:
b. We can now generalize the total probability rule. Consider the following figure showing the two succeeding events, S and E. There are two probabilities for each of the events, the probability of the event happening or the probability of it not happening.
The probability of the final event given that the second event, E is dependent upon the previous event S or S not happening. Therefore for the dependent events, the total probability of the final event (as per the multiplication rule) would be:
However, if the two events were independent of each other, P(E|S) and P(E|SC) would have been equal to P(E). Therefore,
and we know that P(S) + P (SC) = 1
Therefore,
P (E) = P (E)
Hence we can say that P(E) only equals P(E) and nothing else in the case of independent events.