###### Portfolio Returns

A portfolio is basically a collection of investments held by a company, mutual... **Read More**

Probability rules are the concepts and facts that must be taken into account while evaluating the 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.

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 fares increase.

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

For non-mutually exclusive events, the multiplication rule states that:

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

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 \(P(A|B)\) as:

The probability that the first ball to be drawn is blue, P(B) = \(\cfrac {16}{30}\).

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

Thus,

$$ \begin{align*} P(AB) & = P(A | B)P(B) \\ & = \cfrac {15}{29} × \cfrac {16}{30} \\ & =\cfrac {240}{870} \\ & =\cfrac {8}{29} \\ \end{align*} $$

We use the addition rule to assess the probability that events A **or** B occur.

If A and B are mutually exclusive events:

$$ \text P(\text{A or B}) =\text P(\text A) + \text P(\text B) $$

If A and B are non-mutually exclusive events:

$$ \text P(\text{A or B}) =\text P(\text A) + \text P(\text B) – \text P(\text {AB}) $$

This is because we have to remove one of the two instances of the intersection of A and B, as shown in the following figure.

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 0.4, what is the probability of relaxed trade restrictions **or **a price war?

**Solution**

$$ \begin{align} \text P(\text{A or B}) &= \text P(\text A) + \text P(\text B) – \text P(\text {AB}) \\ & = 0.5 + 0.2 – 0.4 = 0.3 \end{align} $$

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:

$$ \text P(\text A) = \text P(\text A | \text B_1)\text P(\text B_1) + \text P(\text A | \text B_3)\text P(\text B_3) + … + \text P(\text A | \text B_n)\text P(\text B_n) $$

Perhaps an example will help you understand the concept:

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?

**Solution**

We simply add the respective areas:

$$ 50 + 65 + 74 = 189 \text{ km}^2 $$

## Question

An analyst analyzed defaults of one hundred and fifty corporate bonds, investment grade and non-investment grade. The following table summarizes the results.

$$ \begin{array}{l|c|c|c}\\ {}&\textbf{Bonds Default}&\textbf{Bonds did not}&\textbf{Total}\\ {}&{}&\textbf{default}&{}\\ \hline \text{Non-investment}&60&15&75\\ \text{grade bonds}&{}&{}&{}\\ \hline \text{Investment grade}&10&65&75\\ \text{bonds}&{}&{}&{}\\ \hline \text{Total}&70&80&150\\ \end{array} $$

If one bond from the sample of 150 bonds is selected at random, determine the probability that the bond defaulted given that the bond was non-investment grade is:

- 46.67%.
- 80.00%.
- 81.23%.

SolutionThe correct answer is

B.The number of bonds that were defaulted and were non-investment grade were 60. The number of bonds that were non-investment grade were 75.

So, the probability that the bond defaulted given that the bond did not default = 60 / 75 = 0.8 = 80%.

A is incorrect.It represents the probability of bonds defaulted out of total sample of 150 bonds (70 / 150).

C is incorrect.It represents the probability that the bond was investment grade given that the bond did not default. The number of bonds that did not default and were investment grade are 65. The number of bonds that did not default were 80. So the probability that the bond was investment grade given that the bond did not default = 65 / 80 = 81.23%.