Probability rules are the concepts and established facts that must be taken into account while evaluating probabilities of various events. The CFA curriculum requires candidates to master 3 main rules of probability. These are the multiplication rule, the addition rule and the law of total probability. We now look at each rule in detail.

**Multiplication Rule**

We use the multiplication rule to determine the **joint probability** of two events, P(AB). A joint probability is the probability of two events happening together. For example, we may be interested in the probability that both gas prices and bus fare increase.

*Note: If two events are mutually exclusive, then they cannot occur together. Therefore, we say that such events have zero joint probability*

The multiplication rule states that:

*P(AB) = P(A | B)P(B)*

**Example**

Let’s assume we have a bag containing 16 blue balls and 14 yellow balls. Suppose we draw two balls at random, one after the other without replacement. Let’s define:

Event B = the first ball is blue

Event A = the second ball is blue

What will be the joint probability of A and B?

**Solution**

From the wording of the question, we can calculate the conditional probability P(A|B):

The probability that the first ball to be drawn is blue, P(B) = 16/30

The probability that the second ball to be drawn is blue given the first one is also blue, P(A | B) = 15/29

Thus, P(AB) = P(A | B)P(B)

= 15/29 * 16/30

= 240/870

= 8/29

**Addition Rule**

We use the addition rule to assess the probability that A or B or both occur

*P(A or B) = P(A) + P(B) – P(AB)*

**Example**

Suppose the probability of relaxed import restrictions is 0.5 and the probability of a price war is 0.2. If the joint probability of relaxed import restrictions and a price war is 0.04, what is the probability of relaxed trade restrictions **or **a price war?

Solution

P(A or B) = 0.5 + 0.2 – 0.04 = 0.66

*Note: If two events are mutually exclusive, then:*

*P(A or B) = P(A) + P(B) since the joint probability would be (and is always) zero.*

**Total Probability Rule**

Suppose we have a set of mutually exclusive and exhaustive events B_{1}, B_{2}, B_{3}…B_{n}

We can determine the unconditional probability of an event, given the conditional probabilities i.e.

P(A) = P(A | B_{1})P(B_{1}) + P(A | B_{3})P(B_{3}) + … + P(A | B_{n})P(B_{n})

Perhaps an example will help you understand the concept:

**Example**

Suppose a local authority subdivides a forest into three regions, B_{1}, B_{2}, and B_{3}. Assume the area covered by each region is 50km^{2}, 65km^{2} and 74km^{2 }respectively. What is the total forest area, A?

The solution is simple: we just add the respective areas i.e.

50 + 65 + 74 = 189 km^{2}

This is the idea behind the law of total probability. The total forest area is replaced by P(A). We add the amount of probability of A that falls in each of the partitions B_{1}, B_{2}, and B_{3}.

*Reading 9 LOS 9e*

*Explain the multiplication, Addition, and total probability rules.*