Unconditional probability using the total probability rule
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:
Compute the unconditional probability of an increase in sales of the company by more than 20 percent.
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.