2.2.1.2 Binomial Distribution

Binomial Distribution is a distribution in which every trial has only two possible outcomes. The distribution has some key characteristics:

• There are two potential outcomes per trial
• The number of trials (denoted by ) is fixed
• The probability of success (denoted by ) is the same for all the trials
• Each trial is independent

The term “Binomial” inherently signifies the presence of two terms. For instance, it can represent outcomes like heads or tails, true or false, or yes or no. In the context of binomial distribution, one outcome is typically labeled as 'success,' while the other is termed 'failure'.

Success and failure are just the terminologies used for describing the outcomes we are looking for in our trial. Say, for a given trial we might be looking for a number of people diagnosed with cancer as a successful outcome (though it doesn't sound like a very successful outcome that we would like to have in reality). These random trials or experiments in which there are only two possible outcomes - success or failure are known as Bernoulli Trial in statistics.

Binomial distribution can be calculated through the following formula:

        

Example 2.3. A study shows that 20% of pregnant women suffer from diabetes during pregnancy, A random sample of 10 women is taken. Find the probability of the following scenarios:

1. All 10 women are diabetic
2. No women are diabetic
3. Exactly 4 women are diabetic
4. At least 4 women are diabetic

As the problem has two outcomes (Diabetic or non-diabetic women), it can be considered a binomial distribution. Here, the values of  and  are the same for all the scenarios:

•  (number of trials) = 10
•  (probability of diabetic women) = 0.2

Scenario 1: All 10 women are diabetic

Here,  (number of successes i.e., diabetic women for whom we want to calculate the probability) = 10

Substituting the values of ,, and  in the binomial distribution formula yields:

        

        

In 0! is always 1. Anything raised to the power of 0 is always 1.

Binomial distribution for this problem can be calculated in excel through the following formula:            =BINOM.DIST(, , , cumulative)            =BINOM.DIST(10, 10, 0.2, FALSE) = 0.0000001024

The formula may differ in older versions of excel. On plotting the same on a chart (Fig 2.2), we get:

The chart can be inferred as follows:

• The x-axis represents the number of successes ()
• The y-axis represents the probability of success () for , depicted by a bar
• The height of the bar represents the probability of those specific number of successes

• The x-axis represents the actual number of successes that occur within a specific interval ()
• The y-axis represents the probability of success () for , depicted by bars.

The chart shows that for 10 random samples of women (), the probability of all 10 women () being diabetic is really low 0.0000001024 or 0.00001024%, which matches with our calculated result.

The following site has been used to draw the probability distribution charts used in this chapter: https://keisan.casio.com/menu/system/000000000540 Play around with the various Probability Distribution plots available on this site.

Scenario 2: No women are diabetic

Here,  (number of successes i.e., diabetic women for whom we want to calculate the probability) = 0

Substituting the values of ,, and  in the binomial distribution formula yields:

        

On plotting the same on a chart (Fig 2.4), we get:

        

The chart shows that for 10 random samples of women (), the probability of 0 women () being diabetic is 0.1073 or 10.73%, which matches with our calculated result.

Scenario 3: Exactly 4 women are diabetic

Here,  (number of successes i.e., diabetic women for whom we want to calculate the probability) = 4

Substituting the values of ,, and  in the Binomial Distribution formula yields:

        

On plotting the same on a chart (Fig 2.5), we get:

        

The chart shows that for 10 random samples of women (), the probability of 4 women () being diabetic is 0.0880 or 8.8%, which matches with our calculated result.

Scenario 4: At least 4 women are diabetic

Here, we want to calculate for (  4). Thus, we have:

 (number of successes i.e., diabetic women for whom we want to calculate the probability) = 4,5,6,7,8,9,10

One way to solve this problem would be to solve for  = 4,5,6,7,8,9,10 and add the probabilities to get a cumulative probability distribution, like this:

        

Or a shorter way could be to solve for  = 0,1,2,3 add them up and subtract from 1, like this:

        

Binomial distribution for this problem can be calculated in excel through the following formula: =1 - BINOM.DIST.RANGE(, , ,) =1 - BINOM.DIST.RANGE(10, 0.2, 0, 3) = 0.1209

However, these approaches are pretty lengthy, so, an alternate approach to solve these kinds of problems is through the Binomial Distribution Table.

The Binomial distribution table gives us a Cumulative Probability Distribution pertaining to .

Say,  = 10 in the table shows the probability distribution for all the values of  = 10 or less, the sum of which is equal to 1.

        

To find the probability distribution for (  4), we can perform the following steps:

• Lookup for cumulative probability distribution for  = 10 by looking for  = 10,  = 10 &  = 0.2 in the table.
• Lookup for cumulative probability distribution for  = 3 by looking for  = 10,  = 3 &  = 0.2 in the table.
• Subtract the cumulative probability distribution of  = 3 from  = 10 to get the cumulative probability distribution of   4

Thus, the probability of getting (  4) is:

        

On plotting the same on a chart (Fig 2.8), we get:

        

Thus, the probability of at least 4 women being diabetic is 12.09%

Fig 2.8 shows that for 10 random samples of women (), the probability of at least 4 women () being diabetic is 0.1209 or 12.09%, which matches with our calculated result.