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(R^C)	0.60	Rise P(SR \| R^C)	0.70
No recession P(R^C)	0.60	Fall P(SR^C \| R^C)	0.30
Recession P(R)	0.40	Rise P(SR \| R)	0.20
Recession P(R)	0.40	Fall P(SR^C \| 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

Create Your Free Account

Create a free account to access this content and join our community of learners.

You'll get access to:

Access the full tutorial
Join our learning community
Track your progress
Bookmark content for later

Learn

Resources

Unconditional Probability Using Total Probability Rule

Create Your Free Account

You'll get access to:

Joint Probability of a Number of Independent Events

Expected Value of Investments

Topics

Course Downloads

Probability Concepts