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Binomial Distribution

HStatistics.com

Binomial Distribution

Before we begin with what binomial distribution is, let's take a few simple questions.

There is a bag which contains 3 Red balls and 2 Blue balls. And suppose you pick up a ball at random.
What would be the probability that the picked ball is red in colour?

So, the probability of a Red ball = 3/5 or, 0.6
and, the probability of a Blue ball = 2/5 or, 0.4

Now, if I ask you to repeat this experiment 5 times.
Meaning, you pick up a ball note it's colour and put it back in the bag.
This is repeated 5 times.
Then, what would be the probability that the first ball was red, second was red, third was blue, fourth was blue and fifth was blue.
ie, what would be the probability that you get the sequence R R B B B?

Since this is an AND event, ie. Red AND Red AND Blue AND Blue AND Blue
The probability would be 0.6*0.6*0.4*0.4*0.4
Or,
0.6^2 * 0.4^3
Now, this was the answer for a specific sequence, ie. R R B B B.

But, if I asked what would be the probability of getting EXACTLY 2 Red Balls in the 5 trials (experiments)?
What would be the answer for that?

To get the answer all we need to do is find the number of different combinations of R R B B B.
and each of those combinations would have the probability = 0.6^2 * 0.4^3
Thus, our answer would be

No. of combinations of (R R B B B) * 0.6^2 * 0.4^3
Or,
5C2 * 0.6^2 * 0.4^3

So, this was the answer for the probability of getting Exactly 2 Red balls in the given 5 experiments.
Similarly, we can find the probability for Exactly 3 Red balls, Exactly 4 Red balls and so on till Exactly N red balls.
The entire distribution would look like:

Exactly 0 Red = 5C0 * 0.6^0 * 0.4^5
Exactly 1 Red = 5C1 * 0.6^1 * 0.4^4
Exactly 2 Red = 5C2 * 0.6^2 * 0.4^3
Exactly 3 Red = 5C3 * 0.6^3 * 0.4^2
Exactly 4 Red = 5C4 * 0.6^4 * 0.4^1
Exactly 5 Red = 5 C 5 * 0.6 ^ 5 * 0.4^0

This distribution is known as Binomial Distribution.

and it's general formula can be put as:
nCr * P^r * (1-P)^(n-r)
Where, P is the probability of Success and N is the number of Trials.

There are 3 important conditions to keep in mind for Binomial Distribution:
1. The number of Trials (N) has to be fixed.
2. The probability of Success (P) should be same for each Trial.
3. There should be only 2 (Bi) possible outcomes of any given trial.