# 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 |

Fall P(SR^{C} | R^{C}) | 0.30 | ||

Recession P(R) | 0.40 | Rise P(SR | R) | 0.20 |

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**

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- Probability - Basic Terminology
- Two Defining Properties of Probability
- Empirical, Subjective and Priori Probability
- State the Probability of an Event as Odds
- Unconditional and Conditional Probabilities
- Multiplication, Addition and Total Probability Rules
- Joint Probability of Two Events
- Probability of Atleast One of the Events Occuring
- Dependent Vs. Independent Events in Probability
- Joint Probability of a Number of Independent Events
- Unconditional Probability Using Total Probability Rule
- Expected Value of Investments
- Calculating Variance and Standard Deviation of Stock Returns
- Conditional Expected Values
- Calculating Covariance and Correlation
- Expected Value of a Portfolio
- Variance and Standard Deviation of a Portfolio
- Bayes’ Theorem
- Multiplication Rule of Counting
- Permutation and Combination Formula

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