# Multiplication, addition and total probability rules

CFA level I / Quantitative Methods: Basic Concepts / Probability Concepts / Multiplication, addition and total probability rules

The multiplication rule is based on the conditional probability.

P(A|B) = P(AB)/P(B), P(B) ≠ 0

where P(AB) = Joint probability of event A and B = Probability that both events A and B occur together

P(A|B) = Conditional probability of event A based on occurrence of event B or probability of event A given B

We know that P(A|B) is the probability of event A given that event B has occurred. Since the event B has already occurred then the probability of A given B will be equal to the probability of occurrence of both events together divided by the probability of occurrence of event B.

Similarly, P(B|A) = P(BA)/P(A) = P(AB)/P(A), P(A) ≠ 0

The addition rule for probabilities helps in calculation of the probability of occurrence of either event A or B.

P(A or B) = P(A) + P(B) - P(AB)

When we add P(A) and P(B), then P(AB) is added twice. So, we subtract it to get P(A or B)

For mutually exclusive events, P(AB) = 0 as those events can never occur together.

Therefore, P(A or B) for mutually exclusive events = P(A) + P(B)

The total probability rule helps in calculation of the unconditional probability of an event. Suppose event A = Probability of increase in stock price. Suppose there are four events that impact the probability of an increase in stock price. Those four events need to be mutually exclusive as well as exhaustive to calculate the total probability of event A. Let's define those four events and their respective unconditional probabilities.

Event A = Increase in stock price

Event A1 = Interest rate < 3 percent
Event A2 = 3 percent ≤ Interest rate < 5 percent
Event A3 = 5 percent ≤ Interest rate < 7 percent
Event A4 = Interest rate ≥ 7 percent

P(A1) = 0.20, P(A2) = 0.25, P(A3) = 0.40, P(A4) = 0.15

Also, P(A|A1) = 0.90, P(A|A2) = 0.70, P(A|A3) = 0.40, P(A|A4) = 0.10

where P(A|A1) is the probability of event A given that event B has occurred and so on.

Now, according to the total probability rule,

P(A) = P(AA1) + P(AA2) + P(AA3) + . .. ..+ P(AAn)

because A will always occur with one of A1, A2,...., An

P(A) = P(A|A1)*P(A1) + P(A|A2)*P(A2) + .... + P(A|An)*P(An)

because P(AA1) = P(A|A1)*P(A1)

So, the probability of event A = 0.90*0.20 + 0.70*0.25 + 0.40*0.40 + 0.10*0.15 = 0.18 + 0.175 + 0.16 + 0.015 = 0.53 = 53 percent.

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