###### Probability in Terms of Odds for and a ...

Odds for and against an event represent a ratio of the desired outcomes... **Read More**

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 S_{1}, S_{2} …S_{n}, then:

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

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:

**Solution**

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

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