Mastering Joint Probability & Total Probability Rule | CFA Level I Quantitative Methods
Conditional and Unconditional Probability
Remember that unconditional probabilities (also known as marginal probabilities) are probabilities of an event happening without any conditions. To illustrate, let’s revisit our stock example:
- Event A: Monthly return (X) ≤ 3%
- Event B: Monthly return (X) > 0%
- Unconditional probability of A: P(A) = 0.5
- Unconditional probability of B: P(B) = 0.7
Now, let’s narrow down the scope to the condition that X > 0. What’s the probability that X ≤ 3%? This is called conditional probability, denoted by P(A|B), which means the probability of A given B.
Given B is true, there’s a 0.29 probability that the return is less than 3% when the return is positive. So, P(A|B) = 0.29.
Joint Probability & the Multiplication Rule
At times, we want to calculate the joint probability, denoted by P(A∩B), which refers to the area of intersection between events A and B.
P(A∩B) = P(A|B) × P(B)
In our example, P(A∩B) = 0.2.
Independent Events & Simplified Multiplication Rule
P(A∩E) = P(A) × P(E)
Lily observes that ABC stock has a 0.6 probability of closing higher each day. What’s the probability of the stock closing higher for 4 consecutive days?
Using the simplified multiplication rule, P(4 consecutive up days) = 0.6 × 0.6 × 0.6 × 0.6 = 0.1296.
Union of Events & the Addition Rule
Sometimes, we want to find the union between events A and B (the probability that at least one of the two events will occur). The addition rule can be used to find this probability:
P(A∪B) = P(A) + P(B) – P(A∩B)
Total Probability Rule
The total probability rule allows us to calculate the probability of an event by considering different scenarios or outcomes. Suppose we have a partition of the sample space into mutually exclusive events E1, E2, …, En. Then for any event A, the total probability rule states:
P(A) = P(A∩E1) + P(A∩E2) + … + P(A∩En)
Using the multiplication rule, this can be rewritten as:
P(A) = P(A|E1) × P(E1) + P(A|E2) × P(E2) + … + P(A|En) × P(En)
A1 is the event of the interest rate remaining unchanged, A2 is the event of a rate cut, and A3 is the event of a rate increase. B is the event of the Dow Jones Industrial Average going up that given month. We are given the conditional probabilities, and we want to find the unconditional probability that the Dow will go up in any given month.
Using the total probability rule:
P(B) = P(B|A1) × P(A1) + P(B|A2) × P(A2) + P(B|A3) × P(A3)
Plug in the known probabilities into the total probability rule, and we get an unconditional probability of 0.55. This means that the likelihood of the Dow going up in any given month, regardless of interest rate movements, is 0.55.
In this lesson, we’ve covered the concepts of conditional probability, joint probability, and the total probability rule. We’ve also explored the multiplication and addition rules for probability calculations. These concepts are essential in various aspects of quantitative methods, including investment decision-making and risk assessment. Keep practicing these concepts, and you’ll soon master the art of probability!