Joint probability

CFA level I / Quantitative Methods: Basic Concepts / Probability Concepts / Joint probability

The joint probability of two events is the probability that the two events occur together. It is denoted by P(AB) for event A and B. The joint probability of two events is equal to zero when the events are mutually exclusive because the two events cannot occur together.

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

Example 1: Using multiplication rule to calculate joint probability

The probability of an increase in stock price is 40 percent if the interest rate rises by more than one percent. The probability of an increase in the interest rate by more than one percent is 30 percent. What is the joint probability that both events (increase in stock price and increase in interest rate by more than one percent) occur together?

Solution:

Let us first name the events.

Event A = Probability of increase in stock price
Event B = Probability of a rise in interest by more than one percent

We are given: P(A|B) = 0.40 and P(B) = 0.30
We are asked to calculate P(AB)

Applying multiplication rule,

P(AB) = P(A|B)*P(B) = 0.40*0.30 = 0.12.

So, the probability that occurrence of both increase in stock price and an increase in interest rate by more than one percent is 0.12.

The independent events are the events that are not dependent on the occurrence of the other event.

For independent events A and B: P(A|B) = P(A)
P(B|A) = P(B)

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

The joint probability of independent events is equal to the product of the unconditional probabilities of those events.

For more than two independent events, P(ABCD) = P(A)*P(B)*P(C)*P(D)

The probability that at least one of two events A and B will occur is given by the additional rule as discussed previously and is given by the following formula.

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

Previous LOS: Multiplication, addition and total probability rules

Next LOS: Dependent and independent events