Quantitative Methods - Joint Probability

Joint probability is defined as the probability of both A and B taking place, and is denoted by P(AB).

Joint probability is not the same as conditional probability, though the two concepts are often confused. Conditional probability assumes that one event has taken place or will take place, and then asks for the probability of the other (A, given B). Joint probability does not have such conditions; it simply asks for the chances of both happening (A and B). In a problem, to help distinguish between the two, look for qualifiers that one event is conditional on the other (conditional) or whether they will happen concurrently (joint).

Probability definitions can find their way into CFA exam questions. Naturally, there may also be questions that test the ability to calculate joint probabilities. Such computations require use of the multiplication rule, which states that the joint probability of A and B is the product of the conditional probability of A given B, times the probability of B. In probability notation:

Formula 2.20
Multiplication rule: P(AB) = P(A | B) * P(B)

Given a conditional probability P(A | B) = 40%, and a probability of B = 60%, the joint probability P(AB) = 0.6*0.4 or 24%, found by applying the multiplication rule.

The Addition Rule
The addition rule is used in situations where the probability of at least one of two given events - A and B - must be found. This probability is equal to the probability of A, plus the probability of B, minus the joint probability of A and B.

Formula 2.21
Addition Rule: P(A or B) = P(A) + P (B) - P(AB)

For example, if the probability of A = 0.4, and the probability of B = 0.45, and the joint probability of both is 0.2, then the probability of either A or B = 0.4 + 0.45 - 0.2 = 0.65.

Remembering to subtract the joint probability P(AB) is often the difficult part of applying this rule. Indeed, if the addition rule is required to solve a probability problem on the exam, you can be sure that the wrong answers will include P(A) + P(B), and P(A)*P(B). Just remember that the addition rule is asking for either A or B, so you don't want to double count. Thus, the probability of both A and B, P(AB), is an intersection and needs to be subtracted to arrive at the correct probability.

Dependent and Independent Events
Two events are independent when the occurrence of one has no effect on the probability that the other will occur. Earlier we established the definition of a conditional probability, or the probability of A given B, P(A | B). If A is completely independent of B, then this conditional probability is the same as the unconditional probability of A. Thus the definition of independent events states that two events - A and B - are independent of each other, if, and only if, P(A | B) = P(A). By the same logic, B would be independent of A if, and only if, P(B | A), which is the probability of B given that A has occurred, is equal to P(B).

Two events are not independent when the conditional probability of A given B is higher or lower than the unconditional probability of A. In this case, A is dependent on B. Likewise, if P(B | A) is greater or less than P(B), we know that B depends on A.

Calculating the Joint Probability of Two or More Independent Events
Recall that for calculating joint probabilities, we use the multiplication rule, stated in probability notation as P(AB) = P(A | B) * P(B). For independent events, we've now established that P(A | B) = P(A), so by substituting P(A) into the equation for P(A | B), we see that for independent events, the multiplication rule is simply the product of the individual probabilities.

Formula 2.22
Multiplication rule, independent events: P(AB) = P(A) * P(B)AB)

Moreover, the rule generalizes for more than two events provided they are all independent of one another, so the joint probability of three events P(ABC) = P(A) * (P(B) * P(C), again assuming independence.

The Total Probability Rule
The total probability rule explains an unconditional probability of an event, in terms of that event's conditional probabilities in a series of mutually exclusive, exhaustive scenarios. For the simplest example, there are two scenarios, S and the complement of S, or SC, and P(S) + P(SC) = 1, given the properties of being mutually exclusive and exhaustive. How do these two scenarios affect event A? P(A | S) and P(A | SC) are the conditional probabilities that event A will occur in scenario S and in scenario SC, respectively. If we know the conditional probabilities, and we know the probability of the two scenarios, we can use the total probability rule formula to find the probability of event A.

Formula 2.23
Total probability rule (two scenarios): P(A) = P(A | S)*P(S) + P(A | SC)*P(SC)

This rule is easiest to remember if you compare the formula to the weighted-mean calculation used to compute rate of return on a portfolio. In that exercise, each asset class had an individual rate of return, weighted by its allocation to compute the overall return. With the total probability rule, each scenario has a conditional probability (i.e. the likelihood of event A, given that scenario), with each conditional probability weighted by the probability of that scenario occurring.

Example: Total Probability
So if we define conditional probabilities of P(A | S) = 0.4, and P(A | SC) = 0.25, and the scenarios P(S) and P(SC) are 0.8 and 0.2 respectively, the probability of event A is:

P(A) = P(A | S)*P(S) + P(A | SC)*P(SC) = (0.4)*(0.8) + (0.25)*(0.2) = 0.37.

The total probability rule applies to three or more scenarios provided they are mutually exclusive and exhaustive. The formula is the sum of all weighted conditional probabilities (weighted by the probability of each scenario occurring).

Using Probability and Conditional Expectations in Making Investment Decisions
Investment decisions involve making future predictions based upon all information that we believe is relevant to our forecast. However, these forecasts are dynamic; they are always subject to change based on new information being made public. In many cases this new information causes us to modify our forecasts and either raise or lower our opinion on an investment. In other words, our expected values are conditional on changing real-world events, and thus can never be perceived as unconditional probabilities. In fact, a random variable's expected value is the weighted average of conditional probabilities, weighted by the probability of each scenario (where scenarios are mutually exclusive and exhaustive). The total probability rule applies to determining expected values.

Expected Value Methodology
An expected value of a random variable is calculated by assigning a probability to each possible outcome and then taking a probability-weighted average of the outcomes.

Example: Expected Value
Assume that an analyst writes a report on a company and, based on the research, assigns the following probabilities to next year's sales:

Scenario Probability Sales ($ Millions)
1 0.10 $16
2 0.30 $15
3 0.30 $14
3 0.30 $13

The analyst's expected value for next year's sales is (0.1)*(16.0) + (0.3)*(15.0) + (0.3)*(14.0) + (0.3)*(13.0) = $14.2 million.

The total probability rule for finding the expected value of variable X is given by E(X) = E(X | S)*P(S) + E(X | SC)*P(SC) for the simplest case: two scenarios, S and SC, that are mutually exclusive and exhaustive. If we refer to them as Scenario 1 and Scenario 2, then E(X | S) is the expected value of X in Scenario 1, and E(X | SC) is the expected value of X in Scenario 2.

Tree Diagram
The total probability rule can be easier to visualize if the information is presented in a tree diagram. Take a case where we have forecasted company sales to be anywhere in a range from $13 to $16 million, based on conditional probabilities.

This company is dependent on the overall economy and on Wal-Mart's same-store sales growth, leading to the conditional probability scenarios demonstrated in figure 2.7 below:

In a good economy, our expected sales would be 25% likely to be $16 million, and 75% likely to be $15 million, depending on Wal-Mart's growth number. In a bad economy, we would be equally likely to generate $13 million if Wal-Mart sales drop more than 2% or $14 million (if the growth number falls between -2% and +1.9%).

Expected sales (good economy) = (0.25)*(16) + (0.75)*(15) = 15.25 million.

Expected sales (bad economy) = (0.5)*(13) + (0.5)*(14) = 13.5 million.

We predict that a good economy is 40% likely, and a bad economy 60% likely, leading to our expected value for sales: (0.4)*(15.25) + (0.6)*(13.5) = 14.2 million.

Advanced Probability Concepts
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