Probability Concepts

Before we learn about the probability concepts, it is important to know the basic terminology.

Random Variable

A random variable is one of the most important concepts in finance. A random variable is a variable whose value is an outcome of a random phenomenon, for example, this can be viewed as the outcome of throwing a die where the process is fixed but the outcome is not.

Random variables describe key things like asset returns. We then use distribution functions to characterize the random variables.

Outcome

An outcome is a possible result of a probability experiment. For example, a portfolio earning 8% returns is an outcome.

An Event 

An event is a single outcome or set of outcomes to which we assign a probability. For example, a portfolio earning 8% returns, or a portfolio earning returns between 6-8%.

Mutually Exclusive Events

These are the events that are mutually exclusive, that is, they cannot happen at the same time. For example, when we throw a die, if one event is to get number 3 and another event is to get number 4 both these events are mutually exclusive; they cannot happen together.

Exhaustive Events

Mutually exhaustive events are those that include all possible outcomes. This means at least one of the events must occur. For example, when we roll a die, the events 1, 2, 3, 4, 5, and 6 are mutually exhaustive, because they include the entire range of possible outcomes. This example is actually mutually exclusive and exhaustive as apart from being all encompassing, no two events can occur together.

Notations

We will use the following notations for probability throughout the reading.
<ul>
 	<li>E denotes an event</li>
 	<li>P(E) denotes the probability of an event</li>
</ul>

Probability - Basic Terminology

There are two important properties of probability.
<ul>
 	<li>The probability of an event E is between 1 and 0, i.e., 0 &lt; P(E) &lt; 1.</li>
 	<li>The sum of probabilities of all mutually exclusive and exhaustive events is equal to 1.</li>
</ul>
These two properties together define probability.

When we roll a die, the events 1, 2, 3, 4, 5, and 6 are mutually exclusive and exhaustive. The probability of any event occurring is between 0 and 1. The sum of probabilities of all these 6 events is equal to 1.

Two Defining Properties of Probability

There are three types of probabilities:
<table>
<tbody>
<tr>
<td width="149">Empirical Probability</td>
<td width="467">
<ul>
 	<li>Based on observed or historical data.</li>
</ul>
</td>
</tr>
<tr>
<td width="149">Subjective Probability</td>
<td width="467">
<ul>
 	<li>Based on an individual’s judgement about the probability of occurrence of an event.</li>
 	<li>The probability of an event is determined by an individual, based on that person’s past experience, personal opinion, and/or analysis of a particular situation.</li>
</ul>
</td>
</tr>
<tr>
<td width="149">Priori Probability</td>
<td width="467">
<ul>
 	<li>Based on prior knowledge.</li>
 	<li>Involves deductive reasoning.</li>
</ul>
</td>
</tr>
</tbody>
</table>

Empirical, Subjective and Priori Probability

The probabilities can also be stated in terms of odds. For example, odds for an event or odds against an event.

Let’s say that the probability of an event occurring is P(E). If we roll a die, the probability of getting a 5 is P(5) = 1/6.

Odds for E

The odds for Event E to occur can be stated as follows:

<img class="aligncenter size-full wp-image-19139" src="https://financetrain.com/wp-content/uploads/odds1.gif" alt="odds1" width="536" height="61">

In our example, the odds for getting a 5 are (1/6)/(1 – 1/6)= 1/5 or one-to-five.

Odds against E

The odds against Event E to occur can be stated as follows:

<img class="aligncenter size-full wp-image-19140" src="https://financetrain.com/wp-content/uploads/odds21.gif" alt="odds2" width="537" height="61">

In our example, the odds against getting a 5 are (1 – 1/6)/(1/6)= 5/1 or 5-to-one.

The concept of odds doesn’t have much relevance in finance and investments. It is more commonly used in betting.

State the Probability of an Event as Odds

Let’s say you are asked the following question:
<blockquote>What is the probability of your portfolio earning a return greater than 10%?</blockquote>
This kind of probability is an unconditional probability as the probability is not dependent on the occurrence of any other event. The event, A, is that the portfolio will earn a return greater than 10%. The probability of such an event will be specified as P(A). The calculation is quite simple. The numerator is the sum of probabilities of all returns being above 10%. Assume this is 0.60. The denominator is 1, the sum of probabilities of all possible returns. The probability P(A) = 0.60/1 = 0.60.

Now, let’s ask another related question:
<blockquote>What is the probability of your portfolio earning a return greater than 10% given that the returns are never below 5%?</blockquote>
Notice that we have added a new condition - given that the returns are never below 5%. Now the probability of portfolio earning returns greater than 10% is not unconditional. It is conditional on another event, B, that is, the returns are never below 5%. Such a probability is called conditional probability, and is expresses as P(A|B), the probability of A given B.

Our calculation will now change.

The numerator will still be the same: the sum of probabilities of all returns being above 10%. We assumed this to be 0.60.

The denominator will now consider Event B as well - the sum of probabilities of all returns being 5% or more. Assume this is 0.80.

The conditional probability will be calculated as P(A|B) = 0.60/0/80 = 0.75.

Unconditional and Conditional Probabilities

<h3>Addition Rule</h3>
The additional rule determines the probability of atleast one of the events occuring.

<img class="aligncenter size-full wp-image-19151" src="https://financetrain.com/wp-content/uploads/rule1.gif" alt="rule1" width="523" height="33">

If A and B are mutually exclusive, then P(A and B) = 0, so the rule can be simplified as follows:

<img class="aligncenter size-full wp-image-19150" src="https://financetrain.com/wp-content/uploads/rule2.gif" alt="rule2" width="536" height="33">
<h3>Multiplication Rule</h3>
Multiplication rule determines the joint probability of two events.

<img class="aligncenter size-full wp-image-19149" src="https://financetrain.com/wp-content/uploads/rule3.gif" alt="rule3" width="526" height="38">

Joint probability of A and B is equal to the probability of A given B multiplied by the probability of B.

If A and B are independent, then P (A/B) = P (A)and the multiplication rule simplifies to:

<img class="aligncenter size-full wp-image-19148" src="https://financetrain.com/wp-content/uploads/rule4.gif" alt="rule4" width="517" height="36">
<h3>Total Probability Rule</h3>
The total probability rule determines the unconditional probability of an event in terms of probabilities conditional on scenarios.

<img class="aligncenter size-full wp-image-19147" src="https://financetrain.com/wp-content/uploads/rule5.gif" alt="rule5" width="536" height="73">

Let’s take an example to understand this.

Event A: Company X’s stock price will rise.

Event B: Inflation will fall. P(B) = 0.6. Therefore, probability of inflation not falling, P(BC) = 0.4

Probability of stock price rising given a fall in inflation, P(A|B) = 0.8

Probability of stock price rising given no fall in inflation, P(A|BC) = 0.6

We can use the total probability rule to calculate the probability of a rise in stock price as follows:

<img class="aligncenter size-full wp-image-19146" src="https://financetrain.com/wp-content/uploads/rule6.gif" alt="rule6" width="505" height="81">

This is the total probability of event A occuring under all scenarios.

Multiplication, Addition and Total Probability Rules

Joint probability refers to the multiplication rule of probability. This is the probability that both the events will occur.

<img class="aligncenter size-full wp-image-19156" src="https://financetrain.com/wp-content/uploads/jp11.gif" alt="Joint Probability" width="540" height="36">

Example

The probability that the price of oil will rise, P(B) = 0.5

The probability that the bus fare will increase if oil price rises, P(A|B) = 0.4

The probability that both oil prices and bus fares will rise, P(AB) = 0.4*0.5 = 0.2

This may look complex but the logic is actually quite straight forward. There is a 50% chance that oil price will rise and if it rises there is a 40% chance that the bus fair will also rise. So, the joint probability of both oil price rise and bus fare rise is 50% of 40%, i.e., 0.5*0.4 = 0.20 or 20%.

Joint Probability of Two Events

This refers to the <a title="Multiplication, Addition and Total Probability Rules" href="https://financetrain.com/multiplication-addition-total-probability-rules/">addition rule</a>.

The additional rule determines the probability of atleast one of the events occuring.

P (A or B) = P (A) + P (B) - P (A and B)

If A and B are mutually exclusive, then P(A and B) = 0, so the rule can be simplified as follows:

P (A or B) = P (A) + P (B) for mutually exclusive events A and B.

Example

An investor is contemplating buying one of the two stocks A or B. The probability P(A) that an investor will buy stock A is 0.30. The probability P(B) that the investor will buy stock B is 0.50. The probability that he may buy both P(A and B) is 0.10.

The probability that the investor will buy atleast one of the two stocks (A, or B, or both) is calculated as follows:

P(A or B) = 0.30 + 0.50 – 0.10 = 0.70

Suppose P(A) and P(B) are mutually exclusive events, that is, the investor will buy only one of the two stocks, then:

P(A or B) = 0.30 + 0.50 = 0.80

Probability of Atleast One of the Events Occuring

Two events are said to be independent if the occurrence of one event is in no way affected by the occurrence of the other event. Suppose we roll a die and receive a 6. The second time we roll the die, its outcome will not be affected by the fact that we received a 6 in the first roll. The outcome of each roll is independent of each other.

To be independent, one of the following conditions must be true:

P(A | B) = P(A) or P(B | A) = P(B)

If the two events are not independent, then they are said to be dependent, that is, the occurrence of one event influences the occurrence of another event.

Dependent Vs. Independent Events in Probability

If two events are independent, then the <a title="Joint Probability of Two Events" href="https://financetrain.com/joint-probability-two-events/">joint probability</a> of these two independent events is calculated as:
P(A and B) = P(A) x P(B)
Example

Suppose we roll two dice. The joint probability of getting a 1 on first die and a 6 on the second die is given as follows:

Probability of getting a 1 on first die, P(A) = 1/6

Probability of getting a 6 on second die, P(B) = 1/6
P(A and B) = 1/6 * 1/6 = 1/36 = 0.0278
The same rule can be applied to calculate the joint probability of any number of independent events. For example, if there are three independent events, A, B and C, their joint probability will be:
P(A and B and C) = P(A) x P(B) x P(C)

Joint Probability of a Number of Independent Events

As we learned earlier, the <a title="Multiplication, Addition and Total Probability Rules" href="https://financetrain.com/multiplication-addition-total-probability-rules/">total probability rule</a> 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.
<table>
<tbody>
<tr>
<td width="169">State of Economy</td>
<td width="173">Probability of Economic State</td>
<td width="137">Stock Performance</td>
<td width="137">Probability</td>
</tr>
<tr>
<td rowspan="2" width="169">No recession P(RC)</td>
<td rowspan="2" width="173">0.60</td>
<td width="137">Rise P(SR | RC)</td>
<td width="137">0.70</td>
</tr>
<tr>
<td width="137">Fall P(SRC | RC)</td>
<td width="137">0.30</td>
</tr>
<tr>
<td rowspan="2" width="169">Recession P(R)</td>
<td rowspan="2" width="173">0.40</td>
<td width="137">Rise P(SR | R)</td>
<td width="137">0.20</td>
</tr>
<tr>
<td width="137">Fall P(SRC | R)</td>
<td width="137">0.80</td>
</tr>
</tbody>
</table>
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 <a title="Joint Probability of Two Events" href="https://financetrain.com/joint-probability-two-events/">joint probability</a> 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

Unconditional Probability Using Total Probability Rule

Expected value is an important concept in investments. An investor will make use of expected value to estimate the expected returns from their portfolio or to assess other factors such as <a title="Common Ratios in Financial Analysis" href="https://financetrain.com/common-ratios-in-financial-analysis/">financial ratios</a>.

We can use a <a title="What is a Random Variable?" href="https://financetrain.com/what-is-a-random-variable/">random variable</a> to describe asset returns. The expected value of a random variable is defined as the weighted average of all possible outcomes of the random variable. The weights are the probabilities of each outcome.

Let’s say we have a random variable X. Its expected value can be represented as follows:

E(X) = P(x1) x1 + P(x2) x2 + ...+ P(xn) xn

Where,
<ul>
 	<li>E(X) is the expected value of the random variable</li>
 	<li>P(xi) is the probability of each observation</li>
 	<li>Xi represents an observed value of a random variable.</li>
</ul>
In terms of investments, expected returns from an asset can be represented as E(R).

Let’s say an investor is analysing the performance of a stock under different states of economy and comes up with the following:
<table width="365">
<tbody>
<tr>
<td width="108">State of Economy</td>
<td width="137">Probability</td>
<td width="120">Return on Stock</td>
</tr>
<tr>
<td width="108">1</td>
<td width="137">0.20</td>
<td width="120">15%</td>
</tr>
<tr>
<td width="108">2</td>
<td width="137">0.20</td>
<td width="120">-5%</td>
</tr>
<tr>
<td width="108">3</td>
<td width="137">0.20</td>
<td width="120">5%</td>
</tr>
<tr>
<td width="108">4</td>
<td width="137">0.20</td>
<td width="120">35%</td>
</tr>
<tr>
<td width="108">5</td>
<td width="137">0.20</td>
<td width="120">25%</td>
</tr>
</tbody>
</table>
The expected returns from this stock can be calculated as follows:

E(R) = 0.20*15%+0.20*(-5%)+0.20*5%+0.20*35%+0.20*25% = 15%

Expected Value of Investments

We can also calculate the <a title="Variance and Standard Deviation" href="https://financetrain.com/variance-standard-deviation/">variance and standard deviation</a> of the stock returns. The variance will be calculated as the weighted sum of the square of differences between each outcome and the expected returns.

<img class="aligncenter size-full wp-image-19191" src="https://financetrain.com/wp-content/uploads/sd11.gif" alt="sd1" width="537" height="109">

The standard deviation will be:

<img class="aligncenter size-full wp-image-19192" src="https://financetrain.com/wp-content/uploads/sd21.gif" alt="sd2" width="517" height="38">

Remember that the units of measuring standard deviation are the same as the units of measuring stock returns, in this case percentage (%).

Calculating Variance and Standard Deviation of Stock Returns

We can use the concept of conditional probabilities to arrive at the conditional expected values. Conditional expected values are conditional based on another event. For example, expected value of a random variable X given scenario S. A practical example would be to find the expected returns from a stock given rising inflation.

The total probability rule can be stated in terms of the expected values as follows:

E(X) = E(X | S1)P(S1) + E(X | S2)P(S2) + … + E(X | Sn)P(Sn)

Where the scenarios S1, S2, …Sn are mutually exclusive and exhaustive.

Example Using Tree Diagram

The following diagram shows the returns from a stock under different inflationary scenarios.

[caption id="attachment_19195" align="aligncenter" width="527"]<img class="wp-image-19195 size-full" src="https://financetrain.com/wp-content/uploads/cev.gif" alt="cev" width="527" height="314"> Conditional Expected Value[/caption]

The diagram shows that the probability of high inflation is 0.70 and the probability of low inflation is 0.30. Given high inflation, the probability of getting a return of 8% is 0.25 and probability of getting a return of 7% is 0.75. Given low inflation, the probability of getting a return of 6% is 0.40 and probability of getting a return of 5% is 0.60.

In the above tree diagram, the values in green are calculated values.

The joint probability of 8% return and high inflation is = 0.25*0.70 = 0.175

The joint probability of 7% return and high inflation is = 0.75*0.70 = 0.525

The joint probability of 6% return and low inflation is = 0.40*0.30 = 0.12

The joint probability of 5% return and low inflation is = 0.60*0.30 = 0.18

The expected return will be calculated as follows:

E(R) = 0.175*8%+0.525*7%+0.12*6%+0.18*5% = 6.695%

Conditional Expected Values

Both <a title="Coefficient of Variation" href="https://financetrain.com/coefficient-variation/">correlation and covariance</a> are indicators of the relationship between two variables. They indicate whether the variables are positively or negatively related, i.e., how they move together. For example, what is the relationship between the performance of gold and S&amp;P 500 index?

Covariance measures the comovement between two variables i.e. the amount by which the two random variables show movement or change together.

The correlation also indicates the degree to which the two variables are related. It’s a translation of covariance into a unit-less measure that we can understand (-1.0 to 1.0). The correlation of the variable with itself is always 1.

Calculating Covariance and Correlation

Covariance is calculated using the following formula:

<img class="aligncenter size-full wp-image-19200" src="https://financetrain.com/wp-content/uploads/cov11.gif" alt="Covariance" width="510" height="45">

Where Ri and Rj represent the returns on two assets, i and j.

The following example illustrates the calculation of expected returns:
<table width="595">
<tbody>
<tr>
<td width="73">State of Economy</td>
<td width="66"></td>
<td width="57">RA</td>
<td width="47">RB</td>
<td width="85">RA-E(RA )</td>
<td width="85">RB-E(RB )</td>
<td width="182">P*( RA-E(RA ))( RB-E(RB ))</td>
</tr>
<tr>
<td width="73">1</td>
<td width="66">10%</td>
<td width="57">-10%</td>
<td width="47">5%</td>
<td width="85">-26.00%</td>
<td width="85">-9.00%</td>
<td width="182">0.234%</td>
</tr>
<tr>
<td width="73">2</td>
<td width="66">30%</td>
<td width="57">15%</td>
<td width="47">12%</td>
<td width="85">-1.00%</td>
<td width="85">-2.00%</td>
<td width="182">0.006%</td>
</tr>
<tr>
<td width="73">3</td>
<td width="66">30%</td>
<td width="57">18%</td>
<td width="47">19%</td>
<td width="85">2.00%</td>
<td width="85">5.00%</td>
<td width="182">0.030%</td>
</tr>
<tr>
<td width="73">4</td>
<td width="66">20%</td>
<td width="57">22%</td>
<td width="47">15%</td>
<td width="85">6.00%</td>
<td width="85">1.00%</td>
<td width="182">0.012%</td>
</tr>
<tr>
<td width="73">5</td>
<td width="66">10%</td>
<td width="57">27%</td>
<td width="47">12%</td>
<td width="85">11.00%</td>
<td width="85">-2.00%</td>
<td width="182">-0.022%</td>
</tr>
<tr>
<td width="57">E(RA) =16%</td>
<td width="47">E(RB)=14%</td>
<td width="182">Cov = 0.0026</td>
</tr>
</tbody>
</table>
Once we have the covariance, we can calculate the correlation as follows:

<img class="aligncenter wp-image-19199 size-full" src="https://financetrain.com/wp-content/uploads/cov21.gif" alt="Correlation" width="519" height="72">

We already have the covariance. The standard deviations of the two returns can be calculated using the formulas learned under section ‘<a title="Variance and Standard Deviation" href="https://financetrain.com/variance-standard-deviation/">Variance and Standard Deviation</a>’.

Std (Ri) = 9.40%

Std (Rj) = 4.17%

Correlation = 0.0026/(9.4%*4.17%) = 0.66

Refer to the <a href="https://financetrain.com/wp-content/uploads/Covariance-Correlation.xlsx">spread sheet Covariance-Correlation.xlsx</a> for detailed calculations.

Covariance Interpreted

In financial markets covariance is positive when the variables show similar behaviour i.e. larger values of one variable correspond to larger values of another variable and the same holds true for smaller values. &nbsp;When the covariance is negative it means the exact opposite i.e. larger values of one variable correspond to smaller values of another variable.

The strength of the linear relationship however cannot be easily interpreted by the magnitude of the calculated value. &nbsp;In order to interpret the strength a related measure called correlation is used.

Correlation Interpreted

The correlation number would always be in the range of -1 to +1. A value of 1 means that the variables always move in the same direction and a value of -1 means the two always move in the opposite direction. &nbsp;In the case where the variables are independent the covariance is zero which means the correlation is also zero. In other words the two variables do not exhibit any movement relative to each other. Any number in between indicates that the one number moves less positively or negatively in relation to changes in another number.

Calculating Covariance and Correlation

We earlier learned about how to calculate the <a title="Expected Value of Investments" href="https://financetrain.com/expected-value-investments/">expected value</a>, <a title="Calculating Variance and Standard Deviation of Stock Returns" href="https://financetrain.com/calculating-variance-standard-deviation-stock-returns/">variance, and standard deviation</a> of a single random variable or an asset.

Portfolio managers will have many assets in their portfolios in different proportions. The portfolio manager will have to therefore calculate the returns on the entire portfolio of assets. The returns on the portfolio are calculated as the weighted average of the returns on all the assets held in the portfolio.

Using the same properties, we can calculate the expected value (returns), variance and standard deviation of a portfolio.
<h5>Expected Value (Expected returns)</h5>
The formula for portfolio returns is presented below:

<img class="aligncenter size-full wp-image-19205" src="https://financetrain.com/wp-content/uploads/ev1.gif" alt="ev1" width="509" height="53">

w represents the weights of each asset, and r represents the returns on the assets. For example, if an asset constitutes 25% of the portfolio, its weight will be 0.25. Note that sum of all the asset weights will be equal to 1, as it will represent 100% of the investment. The returns here are single period returns with same periods for each asset’s returns.

Let’s take an example of a two asset portfolio to understand how portfolio returns are calculated. Let’s say that our portfolio comprises of two assets A and B and has the following details.
<table>
<tbody>
<tr>
<td width="95">&nbsp;</td>
<td width="142">Investment</td>
<td width="104">Returns</td>
</tr>
<tr>
<td width="95">A</td>
<td width="142">25000</td>
<td width="104">10%</td>
</tr>
<tr>
<td width="95">B</td>
<td width="142">75000</td>
<td width="104">6%</td>
</tr>
</tbody>
</table>
The table presents the amount invested in each asset and the returns from each asset. The total amount invested is $100,000. We can calculate the weights for each asset as follows:

wA&nbsp;= 25000/100000 = 0.25

wB&nbsp;= 75000/100000 = 0.75

We can now calculate the portfolio returns as follows:

<img class="aligncenter size-full wp-image-19204" src="https://financetrain.com/wp-content/uploads/ev2.gif" alt="ev2" width="462" height="48">

The same calculation can be extended for multiple assets.

Expected Value of a Portfolio

We learned about how to calculate the <a title="Calculating Variance and Standard Deviation of Stock Returns" href="https://financetrain.com/calculating-variance-standard-deviation-stock-returns/">standard deviation of a single asset</a>. Let’s now look at how to calculate the standard deviation of a portfolio with two or more assets.

The returns of the portfolio were simply the weighted average of returns of all assets in the portfolio. However, the calculation of the risk/standard deviation is not the same. While calculating the variance, we also need to consider the covariance between the assets in the portfolio. If the assets are perfectly correlated, then the simple weighted average of variances will work. However, when we have to account for the covariance, the equation will change.

Covariance&nbsp;reflects the degree to which two securities vary or change together, and is represented as Cov (Ri,Rj). The problem with covariance is that it has no units, and is difficult to compare across assets. Using covariance, we can calculate the correlation between the assets using the following formula:

<img class="aligncenter size-full wp-image-19226" src="https://financetrain.com/wp-content/uploads/vsd1.gif" alt="vsd1" width="519" height="82">

Correlation

After incorporating covariance, the standard deviation of a two-asset portfolio can be calculated as follows:

<img class="aligncenter size-full wp-image-19225" src="https://financetrain.com/wp-content/uploads/vsd2.gif" alt="vsd2" width="528" height="160">

Standard Deviation of a Two Asset Portfolio

In general as the correlation reduces, the risk of the portfolio reduces due to the diversification benefits. Two assets that are perfectly negatively correlated provide the maximum diversification benefit and hence minimize the risk.

Example

Assume we have a portfolio with the following details:

<img class="aligncenter size-full wp-image-19224" src="https://financetrain.com/wp-content/uploads/vsd3.gif" alt="vsd3" width="541" height="141">

The standard deviation can be calculated as follows:

<img class="aligncenter size-full wp-image-19223" src="https://financetrain.com/wp-content/uploads/vsd4.gif" alt="vsd4" width="518" height="79">

This portfolio has an expected return of 16.80% and a portfolio risk of 11.09%.

Variance and Standard Deviation of a Portfolio

Bayes’ Theorem formula, also known as Bayes’ Law, or Bayes’ Rule, is an intuitive idea. We adjust our perspective (the probability set) given new, relevant information. Formally, Bayes’ Theorem helps us move from an unconditional probability (what are the odds the economy will grow?) to a conditional probability (given new evidence, what are the odds the economy will grow?)

Suppose your daughter tells you that her friend is coming home tomorrow. Since you don’t know anything else, there is a 50% chance that the friend is a female. Now she tells you that her friend has long hair. With this additional information there are now more chances that the friend is a female. Bayes’ theorem can be applied in such scenarios to calculate the probability (probability that the friend is a female.)

A simple representation of Bayes’ formula is as follows:

<img class="aligncenter size-full wp-image-19229" src="https://financetrain.com/wp-content/uploads/bt12.gif" alt="bt1" width="568" height="219">

Example 1

The following information is available regarding drug testing.
<ul>
 	<li>0.5% of people are drug users</li>
 	<li>A test has 99% accuracy. 99% of drug users and 99% of non-drug users are correctly identified by it.</li>
</ul>
The problem question is to find the probability of being a drug user if you’ve tested positive?

The solution according to Baye’s theorem is as follows:
<ul>
 	<li>P(pos|user)=0.99 (99% effective at detecting users)</li>
 	<li>p(user)=0.005 the probability of a number of people being drug users</li>
 	<li>p(pos)=0.01*0.995+0.99*0.005 = 0.0149 which is deduced from the following details: 1% chance of non-users, 99.5% of the population, to be tested positive, plus 99% chance of the users, 0.5% of the population, to be tested positive.</li>
 	<li>P(user/pos) = 0.99*0.005/0.0149 = 0.33</li>
</ul>
The answer we arrive at is that there is only a 33% chance that a positive test is correct

In this example some information is available about the proportions of users versus non-users but in practice such information may not be available or determined.

Example 2

A company expects that there is a 5% probability that the economy will expand. Furthermore, there is a 90% probability that the company’s revenue will rise if the economy expands. If the economy does not expand, there is only 40% probability that company’s revenue will rise.

What is the probability that the economy has expanded given that the company’s revenue has risen.

We want to find out:

P (Economy Expansion | Company Revenue Rises) = P (EE | RR)

P(EE | RR) = P (RR | EE) * P(EE)/P(RR)

P (Economy Expansion) = P(EE) = 0.05

P (Revenue Rise | Economy Expansion) = P(RR | EE) = 0.90

P (Revenue Rise) = P(RR) = 0.05*0.90 + 0.95*0.40 = 0.425

P(EE | RR) = 0.90*0.05/0.425 = 0.106

If the company’s revenue has risen, then there is a 10.6% probability that the economy has expanded.

Bayes’ Theorem

Counting problems have to do with counting the total number of outcomes or logical possibilities of something. For example, if we have to flip a coin, we can easily count the number of outcomes. There are only two possible outcomes, either heads or tails. However, as our problem or data set becomes large and complex, so does the total number of possible outcomes. Counting the outcomes one by one may not be possible then and we will have to use some techniques to make our job easy. In this section we will look at a few such techniques.
<h3>Multiplication Rule of Counting</h3>
Problem 1

If there are A ways of doing something and B ways of doing another thing, then the total number of ways to do both the things is = A x B.

For example, assume that your investment process involves two steps. The first step can be done in two ways and the second step can be done in three ways. There are total 2x3 = 6 ways of carrying out both the steps.

Problem 2: Arranging Items in a group. 

Suppose we have a group of 5 people and we want them to stand in a queue. In the queue the first position can be filled in 5 ways. Now position 1 is filled and we have 4 more people left. The second position can be filled in 4 ways. Similarly, third position can be filled in 3 ways and so on. The total no. of ways these 5 positions can be filled is:

= 5 * 4 * 3 * 2 * 1 = 120

If the number of people was n, then this can be written as

n! = n (n-1)(n-2)(n-3)…1

n! is known as factorial.

Solving n factorial using BA II Plus calculator

Suppose you want to calculate 5!. To solve this on your calculator, press 5[2ND]x!.
<h3>Multinomial Formula (General Formula for Labelling)</h3>
The factorial formula above assumed only one group. However, we may have labelling problems with multiple groups. For example, suppose that we have a group of 10 stocks and we want to label four of these stocks as BUY, three stocks as SELL and 3 stocks as HOLD. What is the total number of ways to do this?

This problem can be solved using the general formula for labelling.

<img class="aligncenter size-full wp-image-19244" src="https://financetrain.com/wp-content/uploads/cp1.gif" alt="cp1" width="552" height="96">

We have k different labels, n1, n2,…nk of each type.

In our example,

n1 = BUY = 4

n2 = SELL = 3

n3 = HOLD = 3

So, the 10 stocks can be labelled in number of ways:

<img class="aligncenter size-full wp-image-19245" src="https://financetrain.com/wp-content/uploads/cp2.gif" alt="cp2" width="543" height="56">

Multiplication Rule of Counting

<h3>Combination Formula</h3>
This is a special case of<a title="Multiplication Rule of Counting" href="https://financetrain.com/multiplication-rule-counting/"> multinomial formula</a> where the types of labels k=2. This means that the n objects can be labelled only in two ways and n1 + n2 = n.

For example, suppose we had to label 4 of our 10 stocks as BUY and the remaining 6 as SELL.

So, n=10, n1 = 4 and n2 = 6.

Let’s say n1 = r = 4, in that case n2 can be rewritten as n2 = n – r or 10 – 4 = 6

We can rewrite our formula as follows:

<img class="aligncenter size-full wp-image-19249" src="https://financetrain.com/wp-content/uploads/pcf1.gif" alt="pcf1" width="529" height="58">

This is called the combination formula and is read as n combination r, i.e., how many ways can we select a group of size r from a group of n objects.

<img class="aligncenter size-full wp-image-19252" src="https://financetrain.com/wp-content/uploads/pcf2.gif" alt="pcf2" width="535" height="53">

The combination problems can be solved directly on your BA II Plus calculator using the nCr function.

The Combination formula has its application in binomial trees.
<h3>Permutation Formula</h3>
Note that in combinations, the order in which the objects are listed does not matter, that is A, B is the same as B, A. However, there could be a situation where the order matters. For example, from our group of 10 stocks, we want to select 4 stocks and rank them as No. 1, 2, 3, and 4. To solve this problem, we need to use the permutation formula which accounts for ordering of objects.

<img class="aligncenter size-full wp-image-19251" src="https://financetrain.com/wp-content/uploads/pcf3.gif" alt="pcf3" width="522" height="57">

This is read as the number of permutations of r objects from total n objects.

In our example:

<img class="aligncenter size-full wp-image-19250" src="https://financetrain.com/wp-content/uploads/pcf4.gif" alt="pcf4" width="531" height="58">

There are 5040 ways of selecting 4 objects from a group of 10 objects when ordering of objects is important.

Probability Concepts

Course Resources

Course Quizzes

Lessons

Take the Next Step in Your Data Career