Unconditional Probability Using the Total Probability Rule

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

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

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

Example: Unconditional Probability

Imagine assessing the performance of a stock under different circumstances and coming up with the following probabilities.

$$ \begin{array}{c|c|c|c} {\textbf{State of}} & {\textbf{Probability of}} & {\textbf{Stock}} & {\textbf{Probability}} \\ {\textbf{Economy}} & {\textbf{Economic State}} & {\textbf{Performance}} & {} \\ \hline {} & {} & {\text{Rise } \text P(\text {SR}|\text R^{\text C}) } & { 0.8} \ {\text {No recession } \text P(\text R^{\text C}) } & {0. https://littlescholarsnyc.com/ 7} & { \text{Fall } \text P(\text {SR}^{\text C}|\text R^{\text C}) } & {0.2} \\ \hline {} & {} & {\text{Rise }\text P(\text {SR}|\text R)} & { 0.3} \\ {\text {Recession } \text P(\text R) } & {0.3} & { \text{Fall } \text P(\text {SR}^{\text C}|\text R) } & {0.7} \\ \end{array} $$

How would you go about determining the total probability of a stock rise?

Solution

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

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

Expected Value

The expected value of a random variable is simply the average of the possible outcomes of that variable, taking the probability weights into account. Therefore:

$$ E\left( X \right) =\sum _{ i=1 }^{ n }{ { X }_{ i }P\left( { X }_{ i } \right) } $$

Example: Expected Value

An analyst anticipates the following returns on an asset:

$$ \begin{array}{c|c} {\textbf{Return}} & {\textbf{Probability}} \\ \hline {5\%} & { 65\%} \\ \hline {7\%} & { 25\%} \\ \hline {8\%} & { 10\%} \\ \end{array} $$

$$ \begin{align*} \text{Expected return} & = 0.05 × 0.65 + 0.07 × 0.25 + 0.10 × 0.08 \\ & = 0.0325 + 0.0175 + 0.008 \\ & = 0.058 \\ \end{align*} $$

Calculating Variance from a Probability Model

The variance of a random variable is the sum of the squared deviations from the expected value weighted by respective probabilities. Therefore:

$$ { \sigma }^{ 2 }\left( X \right) =\sum _{ i=1 }^{ n }{ { \left[ { X }_{ i }-E\left( { X } \right) \right] }^{ 2 }P } \left( { X }_{ i } \right) =\left\{ { \left[ X-E\left( { X } \right) \right] }^{ 2 } \right\} $$

Example: Calculating Variance

Using the data from the previous example, we can compute the variance of return:

$$ \begin{align*} { \sigma }^{ 2 }\left( X \right) & =0.65{ (0.05-0.058) }^{ 2 }+0.25{ (0.07-0.058) }^{ 2 }+0.10{ (0.08-0.058) }^{ 2 } \\ & = 0.000126 \\ \end{align*} $$

Variance is not easy to interpret because it has squared units. Therefore, we usually use the standard deviation which has the same units as the expected value. To get the standard deviation, we simply use the square root of variance:

$$ \begin{align*} \text{Standard deviation} & = \sqrt{\text{Variance}} \\ &= \sqrt{0.000126} \\ & =0.01122 \text{ or } 1.12\% \\ \end{align*} $$

Question

You have been given the following data indicating the returns likely to be earned on a stock alongside the corresponding probabilities:

$$ \begin{array}{c|c} {\textbf{Return}} & {\textbf{Probability}} \\ \hline {4\%} & { 40\%} \\ \hline {5\%} & { 25\%} \\ \hline {6\%} & { 35\%} \\ \end{array} $$

The standard deviation of expected returns is closest to:

A. 0.00007475;

B. 0.0495; or

C. 0.008646.

Solution

The correct answer is C.

The first step involves determining the expected return:

$$ \begin{align*} E(X) & = (0.04 × 0.4) + (0.05 × 0.25) + (0.06 × 0.35) \\ & = 0.0495 \\ \end{align*} $$

Next, we must compute the variance of returns:

$$ \begin{align*} { \sigma }^{ 2 }\left( X \right) & =0.4(0.04–0.0495)^{ 2 }+0.25(0.05–0.0495)^{ 2 }+0.35(0.06 – 0.0495)^{ 2 } \\ & = 0.00007475 \\ \end{align*} $$

Lastly, we find the square root of variance to get the standard deviation of expected return:

$$ { \sigma }= \sqrt{0.00007475} = 0.008646 $$

Reading 8 LOS 8h

Calculate and interpret an unconditional probability using the total probability rule.