# The Binomial Distribution

## Summary

• The formula for binomial distribution is as follows: $P(X\quad =\quad x)\quad =\quad { C }_{ x }^{ n }{ p }^{ x }{ q }^{ n-x }$
• We write the binomial distribution as X ~ Bin(n, p)
• E(X) = np
• variance(X) = npq
• Standard deviation = $\sqrt { npq }$

Binomial distribution is a discrete probability distribution. It has four major conditions that we need to keep in mind when dealing with binomial distribution.

There are fixed number of trials in a distribution, known as n.

Each event is an independent event, and the probability of each event is a mutually exclusive event.

Each trial has two outcomes e.g success or failure, heads or tails and pass or fail.

The probability of success for each trial will remain the same. Meaning, total probability of both events is equal to 100% e.g if the probability that there are 0.2 (20%) females students in a class, thus is automatically means that the probability of male students in a class 0.8(80%).

The formula for binomial distribution is as follows:

$P(X\quad =\quad x)\quad =\quad { C }_{ x }^{ n }{ p }^{ x }{ q }^{ n-x }$

Where:

n= number of trials {0, 1, 2, 3….. n}
x = number of success
p = probability of success
q = probability of failure (1- p)

We write the binomial distribution in the form:

X ~ Bin(n, p)

Where:

X ~ Bin means X has a binomial distribution
n = total number of trials, which can be any number greater than 0
p= the probability of success, which can be any number between 0 and 1

We can thus say that the mean the variance and the standard deviation of the binomial distribution can be calculated by using the following formulas.

E(X) = np
variance(X) = npq
Standard deviation = $\sqrt { npq }$

#### Example #1

Q. If a fair coin is tossed, 20 times what is the probability of getting 5 heads.

Solution:

As the question says fair coin, we need to keep in mind that there are 50% chances of heads and 50% chances of tails.

Step #1

Now we will calculate all the values:

n = 20, since the coin was tossed 20 times

x = 5, as x is the number of success, in this question getting 5 heads is considered success

p = similarly the probability of success will be 0.5

q = 1-0.5 = 0.5

Step#2

We will now plug in the values into the formula above.

$P(X\quad =\quad 5)\quad =\quad { C }_{ 5 }^{ 20 }{ 0.5 }^{ 5 }{ 0.5 }^{ 20-5 }\quad =\quad 0.014$

We can hence, conclude by saying, the probability of getting 5 heads by tossing a coin 20 times is 0.014.

#### Example #2

Q. Out of 800 families, with 5 children each, how many families would you expect to have 3 boys.

Solution:

Step #1

We know that:

n = 5
x = 3
p = 0.5
q = 0.5

We will now plug the values into the formula to get:

$P(X\quad =\quad 3)\quad =\quad { C }_{ 3 }^{ 5 }{ 0.5 }^{ 3 }{ 0.5 }^{ 5-3 }\quad =\quad 0.3125$

We can hence conclude by saying,

The probability of having 3 boys is 0.3125

Now we have the probability, but we were asked how many families would you expect to have 3 boys, since the word expect is used in the question. We know that we have to calculate the mean

So we can say:

E(X) = n p

= 800 x 0.3125

= 250

Thus, there are 20 families who have 3 boys.

#### Example #3

Q. About 75% of the population says that they never read books.
What is the probability that in a group of 4 randomly chosen people all read books.

Solution:

The probability of not read = 0.75 = q
The probability that read = 1- 0.75 = 0.25 = p

${ C }_{ 4 }^{ 4 }{ 0.25 }^{ 4 }{ 0.75 }^{ 4-4 }\quad =\quad 0.039$

We have taken, ${ C }_{ 4 }^{ 4 }$ because we have to find the probability that out of 4 all 4 of them read books.