Introduction to Binomial Distribution #1

What is binomial?

The term binomial means consisting of TWO terms. In the world of statistics and probability this means an event that can have only TWO results. These events have to be mutually exclusive, that is if one occurs then the other event cannot occur. So,e examples of mutually exclusive events are given below:

  1. tossing coin for heads / tails
  2. passing or failing a test
  3. an old machine that will work or wont work
  4. finishing your homework or not finishing it
  5. catching a fish or not catching a fish
  6. getting / not getting a job from an interview
  7. winning / not winning lottery
  8. chance of raining / not raining in a day

These are examples of events that are mutually exclusive and mutually exhaustive. This means that P(raining) + P(not raining) = 1. There can be no third outcome.

What is Binomial Distribution?

The binomial distribution is formed when and event is done multiple times and the results are noted. Lets take an example from the above list. The easiest example to start learning is the coin toss. This is because for most of the coins the Probability of getting a heads or tails is equal to 0.5 that is P(heads) + P(tails) = 0.5 + 0.5 = 1.

Now if we start doing the coin toss multiple times and take note we start seeing a pattern that the number of heads and tails stays more or less equal. Over multiple such experiments you will see that very few times you will get multiple heads or multiple tails. and a graphic distribution then forms like below

Binomial Distribution for different probabilities of success
Binomial Distribution for different probabilities of success

The Swiss mathematician Jacob Bernoulli is credited with the discovery of binomial distribution from his work published posthumously in 1713.

binomial distribution formula
binomial distribution formula

This formula may look complicated at first look but we can split it into 3 parts that can simplify it.
The first part is the permutation fraction. n!/k!(n-k)! can be understood as the number of ways we can succeed getting the desired successful trials. For example if we have 5 trails out of which 2 are successful then the number of ways it can happen is in 10 ways as given below. (S = success, f = failure)

  1. ssfff
  2. sfsff
  3. sffsf
  4. sfffs
  5. fssff
  6. fsfsf
  7. fsffs
  8. ffssf
  9. ffsfs
  10. fffss

The second part is pk . This part is the probability of success raised to the power of number of successful trials.

The third part is (1-p)(n-k) . Here we can see that (1-p) is basically the probability of failure. and (n-k) is the number of trials where that outcome was failure.

The formula provides a mass function and probability as the answer and the figure above shows the distribution of the results based on the number of successes and the probability of success.

Why does This Distribution Seem Familiar?

The distribution image above shows an uncanny resemblance to the Normal Distribution. That is because binomial distribution having a sufficiently large number of trials can be approximated as a normal distribution.
The Galton Board Experiment is a great example where Binomial distribution with a large number of trials tends to look like a Normal distribution. This is due to the Central limit theorem, which tells us that for large sample sizes the mean will be more or less equal to the mean of the population of the normal distribution with similar finite variance irrespective of the distribution.

Practice Problems

Let us solve a few problems with binomial distribution to become familiar with the topic.
Suppose you went on a fishing trip to a large lake where the probability of catching a fish is 0.4 when the line is cast once.

  1. what probability of 8 fish caught with 15 casts
  2. what is probability that 12 fish are caught in 20 casts
  3. what is probability that less than 5 fish are caught with 10 casts
  4. what is probability that more than 9 fish are caught with 12 casts
  5. what is probability that 7 fish are caught with 25 casts

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