Calculating Unconditional Probability Using Total Probability

We can use the total probability rule to determine the unconditional probability of an event in terms of conditional probabilities in certain scenarios.

In general, if we have a set of mutually exclusive and exhaustive events S₁, S₂ …S_n, then:

$$ \text P(\text T) = \text P(\text T | \text S_1) \text P( \text S_1) + \text P(\text T | \text S_2) \text P(\text S_2) + … + \text P(\text T | \text S_{n}) \text P(\text S_n) $$

Question

Suppose you assess the performance of a stock under different circumstances and come up with the following probabilities.

$$ \begin{array}{c|c|c|c} {\textbf{State of}} & {\textbf{Probability of}} & {\textbf{Stock}} & {\textbf{Probability}} \\ {\textbf{Economy}} & {\textbf{Economic State}} & {\textbf{Performance}} & {} \\ \hline {} & {} & {\text{Rise } \text P(\text {SR}|\text R^{\text C}) } & { 0.8} \\ {\text {No recession } \text P(\text R^{\text C}) } & {0.7} & { \text{Fall } \text P(\text {SR}^{\text C}|\text R^{\text C}) } & {0.2} \\ \hline {} & {} & {\text{Rise }\text P(\text {SR}|\text R)} & { 0.3} \\ {\text {Recession } \text P(\text R) } & {0.3} & { \text{Fall } \text P(\text {SR}^{\text C}|\text R) } & {0.7} \\ \end{array} $$

The total probability of a stock rise is closest to:

1. 0.50.
2. 0.65.
3. 0.70.

Solution

The correct answer is B.

You need to find the unconditional probability of a stock rise under all circumstances. Therefore,

$$ \begin{align*} \text P(\text{SR}) & = \text P(\text {SR} | \text R^\text{C}) \text P(\text R^\text{C}) + \text P(\text{SR} | \text R) \text P(\text R) \\ & = 0.8 × 0.7 + 0.3 × 0.3 \\ & = 0.65 \\ \end{align*} $$