The Binomial Distribution: A Comprehensive Explanation

Binomial Distribution is a group of cases or events where the result of them are only two possibilities or outcomes. The good and the bad, win or lose, white or black, live or die, etc.

For example, when the baby born, gender is male or female. When we are playing badminton, there are only two possibilities, win or lose. When you answer a question, it’s just right or wrong answer.

Binomial is a case of a random variable. It means the group of experiments should be independent and not affected by others.

When we just have two options, we can use the binomial distribution to solve it.

Characteristics of binominal distribution

A random variable has a binomial distribution if met this following conditions :

1.  There are fixed numbers of trials (n).

2.  Every trial only has two possible results: success or failure.

3. The probability of success for each trial is always equal. Usually, the success one symbolized with (p).

4. The trials are independent. It means the first trial can not influence any other trial.

Let’s take a look in this example :

You have 8 times a trial of throwing dice into the table. You want to count the number of even numbers showed up. Does this representation a binomial random variable? Let’s take a look at the following lists below!

1. There are 8 times of trial and it means we have numbers of fixed trials. Characteristic 1 is met. The n = 8.

2. The outcome of each throw is even or odd. It means, there are only two possible results. Even or odd. In this case, success means even number showed up and failure means odd number showed up. Characteristic 2 is met.

3. The probability for each possible outcome is equal. As we know, the probability of success (even number showed up) is 50 percent or 0,5. It means the probability of failure (odd number showed up) is also 0,5. Characteristic 3 is met.

4. We assume the dice are being thrown in the same way. It means, the first thrown does not affect the second, and so on. Characteristic 4 is met.

Well, we can conclude that the example can be processed by using the binomial distribution formula. All the characteristic is fulfilled.

Binomial Distribution Formula

Okay, let’s get started to the formula. Lucky, we just have to use the existing formula that already proven. There is no obligation for us to prove it.

binomial-distribution-formula

n = fixed numbers of trials

x = specified numbers of success trials

n-x = numbers of failures trial

p = probability of success on trial

q = probability failure on trial (some textbook use letter 1-p to denote it)

Example :

Suppose we are playing dice. You have the chance to throw the dice 8 times. You bet that number “3” will appear in these games. Find the probability!

Answer :

Now, let’s define it slowly and clearly.

example-of-binomial-distribution

Conclusion: the probability of number “3” dice showed up in 8 times trial is 0.11.

Finding probabilities using the binomial table table

The fact is, tons of binomial probabilities are already provided. You can find it on this binomial table. But first, we have to understand and make sure how to use the table. It called the binomial table. Each table has rows and columns. You see many possible values for many experiments.

1. Finding probabilities value if p ≤ 0.50

Use the table on the link above and look at it carefully. Read the steps below :

1. Find the right binomial table with right of total numbers (n)

2. Find the column that represents your probability value (p) or the closest one.

3. Find the row that represents the number of successful events (x) you are interested in.

4. See the selected row and column. That is the probability for x successes.

Let us use it in the example.

Suppose you are a manager and get a report that the average of the damaged product of the year is about 10 percent. If you are going to take 5 products sample,  find the probability that 1 of the product is damaged!

Now, let’s define it slowly and clearly.

x = 1

n = 5

p = 0.1

q = 0.9

another-example-of-binomial-distribution

Now, let’s see the table on binomial table.

result-of-binomial-distribution-by-using-binomial-table

See? You’ll the same exact number with the formula calculation. This is not too difficult, is not it?

2. Finding probabilities value if p > 0.50

This is a bit tricky. Binomials probabilities table shows us the probability of success for several different n and p values. This is because it’s still possible to use the prepared table.

To use the same table on the appendix page for p > 0.50, read the steps below :

1. Find the right binomial table with right of total numbers (n)

2. Instead of looking at the probability column of success, find the column that represents 1 – p (probability of failure).

3. Find the row that represents the number of failure events (n-x) that are related to the success value (x) you wanted.

4. See the selected row and column. That is the probability of the number of failures. It also the same value for the number of successes events (x) that you wanted.

5. There is no need to take the compliment to summarize the final answer. The complements are already counted by using 1-p instead of success probability.

Actually, it’s just a reverse thinking. We reverse the probability into the smaller one use the opposites of what we are looking for.

Let us use the sample example above, but now, we are calculated the probability of damaged products taken.

You are a manager and get a report that the average of the damaged product of the year is about 0.6 percent. If you are going to take 5 products sample, find the probability that 1 of the product is damaged!

Now, let’s define it slowly and clearly.

x = 1 (number of successes event)

n = 5 (numbers of events)

p =  (probability of damaged products successes)

q = 0.4 (probability of good products successes)

how-to-use-binomial-table

We can conclude that the probability of one damaged product is 0.077.

Now, let us try using the binomial table.

As I said before, it’s just about thinking back. If the chance for success is 0.6, then it will be rotated to 0.4. If the chance for success is one damaged product, it is now turned into 4 products in good condition. All numbers of events are still the same, 5.

Let’s define it!

x = 4 (number of success event by reverse it to the good products)

n = 5 (number of events)

p =  (probability of good products successes)

q = 0.6 (probability of damaged product successes)

Now, look at binomial table for p=0,40; n=5, x=4;. You’ll see the exact value as calculated above. Cheers!

Finding probabilities for X greater than, less than, or between two values.

The binomial table shows probabilities for X to a specific value. To find probabilities for x being greater than, or less than, or between two values, just find the correspondent values in the binomial table and add their probabilities. It’s just like logic math.

For example, the director gets a report that the average of the damaged product of the year is about 0.3 percent. If you are going to take 5 products sample, find the probability that more than 2 products are damaged!

X =  x > 2

n = 5

p = 0.3

q = 0.7

Answer :

P (X >2) = P(X=3) + P(X=4) + P(X=5)

                = 0.1232 + 0.028 + 0.002

                = 0.162 (look at the binomial table)

Conclusion : the probability of more than  products are damaged is 0.162

The expected value (mean) and variance of the binomial distribution

The mean of the random variable is the average of all possible values over the populations or individual. It’s calculated by multiplying the weighted average of x values with their probabilities.

mean-of-binomial-distribution

The variance of the random variable is the weighted average of the squared deviation from the mean (expected value). If you want to count the standard deviation, just square root the variance

variance-of-binomial-distribution

Example: You flip a coin to the air ten times. X is the number of tails. The probability of tail is 0.5. Summarize the mean and standard deviation.

Answer :

mean-calculation-of-binomial-distribution
how-to-calculate-variance-of-binomial-distribution

Conclusion, the mean is 5 and the standard deviation is 0.5

Mean and variance is basic of inferential statistics that you need to know. You have to put those on your mind because sometimes you might need it.

Closing

There are many types of normal distribution uses in real life. The binomial distribution is just one form of distribution that we often experience in life. By understanding it, we will understand other distributions better.

Do not hesitate to leave a comment!