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 Economy | Probability of Economic State | Stock Performance | Probability |
| No recession P(RC) | 0.60 | Rise P(SR | RC) | 0.70 |
| Fall P(SRC | RC) | 0.30 | ||
| Recession P(R) | 0.40 | Rise 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