# Binomial Experiment

How to find the probability of the binomial experiments: formula, examples, and their solutions.

## Formula

Think about an independent event

whose probability of 'want' is *p*.

(The probability of 'not want' is 1 - *p* = *q*.)

Probability of (not *A*, Complement)

For *n* trials,

if the 'want' happens *k* times,

('not want' happens '*n* - *k*' times)

then that probability is _{n}C_{k}⋅*p*^{k}⋅*q*^{n - k}.

It's like finding the ways of

choosing *p* *k* times

and choosing *q* '*n* - *k*' times.

Binomial theorem

Each probability is multiplied (*p* or *q*)

because each trial is an independent event.

## Example 1

Tossing a coin is an independent event.

And it happens 6 times.

So this is a binomial experiment.*n* = 6 (6 trials)*k* = 3 (3 'head's)*p* = 1/2 (Probability of 'head' shown)*q* = 1 - 1/2 = 1/2 (Probability of 'head' not shown)

Use the binomial experiment formula.

P = _{6}C_{3}⋅(1/2)^{3}⋅(1/2)^{6 - 3}

Cancel 6 and 2⋅3.

Cancel 4 and change 2^{3} to 2.

## Example 2

Spinning a spinner is an independent event.

And it happens 5 times.

So this is a binomial experiment.*n* = 5 (5 trials)*k* = 2 (2 'blue's)*p* = 1/3 (Probability of the 'blue' part)*q* = 1 - 1/3 = 2/3 (Probability of the other part)

Use the binomial experiment formula.

P = _{5}C_{2}⋅(1/3)^{2}⋅(2/3)^{5 - 2}

## Example 3

Tossing a fair die is an independent event.

And it happens 3 times.

So this is a binomial experiment.*n* = 3 (3 trials)*k* = 0, 1 (0 or 1 '6')*p* = 1/6 (Probability of '6' shown)*q* = 1 - 1/6 = 5/6 (Probability of '6' not shown)

Use the binomial experiment formula.

P = _{3}C_{0}⋅(1/6)^{0}⋅(5/6)^{3 - 0} [P(zero '6')]

+ _{3}C_{1}⋅(1/6)^{1}⋅(5/6)^{3 - 1} [P(one '6')]

Find the prime factorizations of 200 and 216.

Then cancel the common factors.