Binomial Probability Formula: Understanding Bernoulli Trials and Probability

If you have ever taken a class in statistics or probability, you have likely run into the concept of binomial probability (even if you didn’t know it by that name). This is one of the most fundamental concepts in probability and finds extensive use in statistics analyzing stock prices and valuing options. This tutorial will give you a basic grounding in binomial probability and teach you how to use the binomial probability formula. For more in-depth tutorials on a numerical approach to valuing options using binomial probability, check out this course on call and put options.

Binomial Probability Formula and Bernoulli Trials

The binomial probability formula is a simple formula for calculating the probability in Bernoulli trials. A thorough understanding of Bernoulli trials is crucial to understanding how binomial probability works and how to calculate it.

Named after famed 18th century Swiss mathematician Daniel Bernoulli, a Bernoulli trial describes any random experiment that has exactly two outcomes – a failure, and a success. The experiment is completely independent, i.e. the probability of failure or success is the same every time you conduct the experiment.

Since a Bernoulli trial has only two outcomes, it can usually be framed as a question with “yes” or “no” answers.

Keep in mind that the terms “failure” and “success” here are used only to denote the possibility of an event happening and not for their literal meanings. That is to say, they carry no value judgments whatsoever.

Explaining Bernoulli Trials with an Example

The best way to understand Bernoulli trials is with the classic coin toss example.

Every time you toss a coin, you have an equal probability of the coin landing either heads or tails (since this is a mathematical exercise, we won’t consider the chance of the coin landing on its edge!). That is to say, there is 50% chance of getting either heads or tails. This holds true regardless of how many times the coin is tossed. You can toss the coin a thousand times and the probability of landing on either side would be still 50%.

If we were to conduct an experiment where heads = success and tails = failure, then we can say that this Bernoulli trial has a 50% success rate, 50% failure rate. Since we classified “heads” as “success”, we can frame this Bernoulli trial as a question – “Did the coin land heads?” Answering “yes” here would mean success, while “no” would imply failure (i.e. you got tails).

In fact, any situation with a yes/no response can be classified as a Bernoulli trial. A survey of voters that asks, “Did you vote in the elections” is a Bernoulli trial. A roll of dice experiment where a number above 4 is “success” is also a Bernoulli trial answered by the question “did you get four or above on your dice?”

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A Mathematical Definition of Bernoulli Trials

Based on the above, we can say that an experiment may be called a Bernoulli trial when it meets the following conditions:

• The number of trials is fixed, not infinite.
• Each trial (for example, each coin toss) is completely independent of the results of the previous turn.
• There can be only two outcomes – success/yes and failure/no.
• The probability of either outcome remains constant from trial to trial. For example, the probability of landing heads in a coin toss remains 50% regardless of what happened in a previous coin toss.

Mathematically, if we say that the probability of success in a Bernoulli trial is p, then the probability of failure in the same trial, q, can be written as:

q = 1 – p

Thus, in a coin toss experiment, if probability of landing heads is 50% or 0.5, then probability of landing tails is:

q = 1 – 0.50

q = 0.50

Similarly, let’s consider a dice roll experiment where we consider landing a 6 to be “success” and anything below that to be “failure”.

We know that probability is defined as:

Probability of an event  = Number of positive outcomes

Total number of outcomes

Here, number of positive outcomes is 1 and total number of possible outcomes is 6 (since there are six number of a dice). Thus, probability of success p (landing a 6) is 1/6.

Based on the above, the probability of failure q can be written as:

q = 1 – 1/6

q = 5/6

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Using Binomial Probability Formula to Calculate Probability for Bernoulli Trials

The binomial probability formula is used to calculate the probability of the success of an event in a Bernoulli trial. Hence, the first thing we need to define is what actually constitutes a success in an experiment. This is completely arbitrary and depends on the experiment itself. One experiment may define it as the chances of rolling a 6 on a dice, another may define it as the chances of rolling 3 or more.

Based on this, we have the following formula:

Probability of k successes in n trials (P) = knCpkqn-k

Where:

n = total number of trials

k = total number of successes

n – k = total number of failures

p = probability of success in one trial

q = probability of failure in one trial (i.e. 1 – p)

knC = n!k!(n-k)! = binomial coefficient

 

An example will illustrate this formula better:

Example: Calculate the probability of rolling 4 on a dice exactly 5 times in 25 trials.

Here, we have the following:

n = total trials = 25

k = total successes = 5

n – k = total failures = 20

p = 1/6 = 0.167

q = 5/6 = 0.833

knC = 25!5!(20)! = 53130

 

Therefore, probability (P) will be:

P     = 53130 x(0.167)5(0.833)20

= 53130 x (0.0001298)(0.02587)

= 53130 x (0.00000335872)

= 0.17844

 

Thus, the probability is 0.17844. This way, we can calculate the probability of any event provided we know the number of trials and the probability of the event occurring in a single trial.

A thorough understanding of probability, especially binomial probability, is a valuable skill when it comes to options pricing. You can learn more about it in this course on advanced options concepts, including probability, Greeks and simulation.