Unconditional probability using the total probability rule

CFA level I / Quantitative Methods: Basic Concepts / Probability Concepts / Unconditional probability using the total probability rule

We have already discussed the calculation of the unconditional probability of an event using the total probability rules.

If the event A is dependent on event A₁, A₂,...., A_n where A₁, A₂,...., A_n are mutually exclusive and exhaustive events then according to the total probability rule:

P(A) = P(AA₁) + P(AA₂) + P(AA₃) + . .. ..+ P(AA_n)

because A will always occur with one of A₁, A₂,...., A_n

P(A) = P(A|A₁)*P(A₁) + P(A|A₂)*P(A₂) + .... + P(A|A_n)*P(A_n)

because P(AA₁) = P(A|A₁)*P(A₁)

So, the unconditional probability of an event is equal to the sum of joint probabilities of the event with mutually exclusive and exhaustive events.

The unconditional probability of an event A is equal to the sum of the product of conditional probabilities of event A with different mutually exclusive and exhaustive events and the probabilities of those events.

P(A) = Conditional probability of event A with event A₁*Probability of event A₁+ ..... + Conditional probability of event A with event A_n*Probability of event A_n

where A₁, A₂,...., A_n are mutually exclusive and independent events.

Example 2: Calculation of unconditional probability

The probability of an increase in sales of a company (based in the USA) by more than 20 percent is dependent on three mutually exclusive and exhaustive events. Those three events are- increase in overall GDP of the USA less than 5 percent, increase in GDP by more than or equal to 5 percent and less than 7 percent, and increase in GDP by more than or equal to 7 percent. The conditional and unconditional probabilities of the events are given below:

Event	Probability	Probability of increase in sales of the company by more than 20 percent
Increase in USA GDP by less than 5 percent	0.40	0.10
Increase in USA GDP by greater than or equal to 5 percent and less than 7 percent	0.35	0.40
Increase in USA GPD by greater than or equal to 7 percent	0.25	0.70

Compute the unconditional probability of an increase in sales of the company by more than 20 percent.

Solution:

Let us first name the events:

Event A= Increase in sales of the company by more than 20 percent

Event A₁= Increase in USA GDP < 5 percent

Event A₂ = 5 percent ≤ Increase in USA GDP < 7 percent

Event A₃ = Increase in USA GDP ≥ 7 percent

We are given the following probabilities:

P(A₁) = 0.40, P(A₂)= 0.35, P(A₃) = 0.25

P(A|A₁) = 0.10, P(A|A₂) = 0.40, P(A|A₃) = 0.70

Applying the total probability rule, we get the unconditional probability of event A:

P(A) = P(A|A₁)*P(A₁) + P(A|A₂)*P(A₂) + P(A|A₃)*P(A₃) = 0.10*0.40 + 0.40*0.35 + 0.70*0.25 = 0.04 + 0.14 + 0.175 = 0.355.

So, the unconditional probability of an increase in the sales of the company by more than 20 percent is 0.355.

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