  Probability distribution is a statistical technique that is used very often by fund managers and stock brokers. It helps them decide if a stock is worth investing in and the range of returns a stock may provide. The history of stock (the returns it provided) over a given time period is used to calculate these predictions. The greater the amount of data available (that is, the longer the history of the stock’s returns stretches) the more accurate your calculations will be. In probability distribution, and increase in the sample size corresponds directly to a reduction in sampling erro (if you really want to know how it’s done – check out this beginners guide to technical analysis of stock charts).

In this tutorial, we’re going to look at the basics of probability distribution. We’ll also create  basic probability distribution tables and explain a couple of advanced concepts.  If you prefer a more visual learning, you can also learn about probability concepts and techniques with this amazing course.

## What is Probability?

So what is probability exactly, in the statistical sense? It is the branch of mathematics that deals with finding the likelihood of the occurrence of an event. It can be represented by a number that has a value between 0 and 1. When the value of closer to 0, it’s unlikely that an event will occur and when the value is closer to 1, it’s almost certain an event will occur.  Learn more about probability, in layman’s terms, with this course.

For example, the probability that the sun will rise tomorrow is 1. The probability that tomorrow is a holiday, on the other hand, is close to 0.  To understand probability distribution better, you need to understand the basic terminology. Lets take a quick look at the commonly used terms:

• Variable: A variable is the occurrence we’re tracking. It is represented by a symbol (a, z, y, etc.) and has a certain mathematical value.
• Random Variable: When the variable is derived as a result of a statistical function, it is considered to be a random variable (unverified but probable). A random variable is usually represented by an uppercase letter.

For example, a random variable can be denoted by an uppercase Z. The probability its occurrence is denoted by P (Z). Suppose we had another variable x. The probability that the value of random variable Z is equal to the other variable x (can also stand for a value) is given by:

`P (Z=x)`

If we know for a fact that the value of Z is in fact equal to the value of x, we can denote this by:

`P (Z=x) = 1`

## What is Probability Distribution?

Probability distribution is a statistical derivation (table or equation) that shows you all the possible values a random variable can acquire in a range. This result (all possible values) is derived by analyzing previous behavior of the random variable. The result can be plotted on a graph between 0 and a maximum statistical value. Where the values lie exactly is influenced by a number of factors, like the skew, distribution mean and standard deviation.

A probability distribution table is nothing but the graphical representation of the probability that a random variable would occur within a specified range. This course can help you build a good foundation to understanding probability distribution.

We’ll take a look at some examples to help you understand the concept better:

## Example 1

Suppose you had a coin with you, with two sides: Heads and Tails. Now, when you flip the coin once, you will either get heads or tails. We can denote that by H and T. If you flip the coin two times, you will get one of four possible occurrence: the coin will show heads twice (HH), the coin will show heads and tails (HT), the coin will show tails first the heads second (TH) or it will show 2 tails (TT). If you flip the coin three times, you will get the following possible outcomes:

`TTT, TTH, THT, THH, HTT, HTH, HHT, HHH.`

Now, let’s say that the variable x shows the number of times you get tails (T) after flipping the coin 3 times. You will get tails either 0 times, 1 time, 2 times or 3 times. The value of x will therefore be within 0 to 3. However, the probability of there being tails just once: P (Z= 1) will be 3/8. This means that 3 times out of 8 you will get just 1 tails( and 3 heads) when you toss a coin thrice.

Now, a probability distribution table can be prepared for this as follows:

 Number of Tails x Probability of Z Being Equal to the values of x -> Result 0 P (Z=0) 1/8 1 P (Z=1) 3/8 2 P (Z=2) 3/8 3 P (Z=3) 1/8

## Example 2

Suppose you have a die with you and you roll it once. There are six possible values it can take: 1, 2, 3, 4, 5 and 6. Now let’s say that the variable x shows how many 2s you can get when you roll the die one time. You will get a 2 one out 6 times you roll the die: P (Z=1) will be 1/6. The rest of the time, you won’t be a getting a 2: P (Z=0) will be 5/6.

Now, we can prepare a probability distribution table for this as follows:

 Number of 2s on Rolling the Die Once Probability of Z Being Equal to the Value of x -> Result 0 P (Z=0) 5/6 1 P (Z=1) 1/6

Notice that if you total the result column, it always comes out to 1. If it’s more or less than one, you went wrong somewhere in your calculations. We recommend you prepare your own probability distribution tables to understand the concept better.

## Cumulative Probability Distribution

What is cumulative probability distribution? As the name suggests, it’s a cumulative property and is obtained by analyzing two or more events. It helps us find the probability of a random variable falling within a certain range.

Continuing with our earlier example, let’s suppose you flip a coin three times. What is the probability that tails occurs 2 times or less? This means that tails can occur 2 times (TTH THT HTT), 1 times (HHT HTH THH) or 0 times (HHH)- and it can’t occur all three times (TTT). Mathematically, we can represent this by:

`P (Z<=2) = P (Z=2) + P (Z=1) + P (Z=0) = 3/8 + 3/8 + 1/8 = 7/8.`

You can represent this in a table, and it would become a cumulative probability distribution table. Why don’t you try preparing one now? Just add another column for cumulative probability distribution, with the following values:

`P (Z<=0), P (Z<=1), P (Z<=2) and P (Z<=3)`

## Probability Distribution: Discrete and Continuous

Probability can either be discrete or continuous. If the variables are discrete and we were to make a table, it would be a discrete probability distribution table. If the variables are continuous, we would obtain a continuous probability distribution table.

What are discrete and continuous variables exactly? Let’s take a look:

• Continuous Variable: When the value of a variable can be absolutely any value in a specified range, it is a continuous variable. For example, a person can be 5 feet tall or 7 feet tall, and absolutely any height in between.
• Continuous Variable: If a variable is not continuous, it is discrete (an integer value). For example, a pizza can have 6, 8 or 12 slices, but it can’t have 7.5 slices.

To learn more about probability, variables and data analysis, you can take this workshop in probability and statistics.  Once you’re ready to move on to the next level, this course can show you how to handle advanced options trading using statistical analysis tools.

Page Last Updated: February 2020