Unconditional Probability Using Total Probability Rule

As we learned earlier, the total probability rule determines the unconditional probability of an event in terms of probabilities conditional on scenarios.

P(A) = P(A | S1)P(S1) + P(A | S2)P(S2) + … + P(A | Sn)P(Sn)

Where the scenarios S1, S2, …Sn are mutually exclusive and exhaustive.

Let’s take one more example of the Total Probability Rule.

An analyst is assessing the performance of a stock under different scenarios. He comes up with the following probabilities.

State of EconomyProbability of Economic StateStock PerformanceProbability
No recession P(RC)0.60Rise P(SR | RC)0.70
Fall P(SRC | RC)0.30
Recession P(R)0.40Rise P(SR | R)0.20
Fall P(SRC | R)0.80

Question 1

Based on the above data, what is the total probability of a stock rise? We need to find the unconditional probability of a stock rise under all scenarios.

P(SR) = P(SR | RC) P(RC) + P(SR | R) P(R)

\= 0.70*0.60 + 0.20*0.40 = 0.5

Question 2

What is the joint probability of having a recession and at the same time having a stock price fall?

P(R and SRC) = P(SRC | R)x P(R) = 0.8*0.4 = 0.32

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