# Binomial Distribution Formula

The binomial distribution is a commonly used discrete distribution in statistics. The normal distribution as opposed to a binomial distribution is a continuous distribution. The binomial distribution represents the probability for 'x' successes of an experiment in 'n' trials, given a success probability 'p' for each trial at the experiment.

**Binomial Distribution in Statistics: **The binomial distribution forms the base for the famous binomial test of statistical importance. A test that has a single outcome such as success/failure is also called a Bernoulli trial or Bernoulli experiment, and a series of outcomes is called a Bernoulli process. Consider an experiment where each time a question is asked for a yes/no with a series of n experiments. Then in the binomial probability distribution, the boolean-valued outcome the success/yes/true/one is represented with probability p and the failure/no/false/zero with probability q (q = 1 − p). In a single experiment when n = 1, the binomial distribution is called a Bernoulli distribution.

## What Is the Binomial Distribution Formula?

The binomial distribution formula is for any random variable X, given by; P(x:n,p) = ^{n}C\(_x\) p^{x }(1-p)^{n-x} **Or** P(x:n,p) = ^{n}C_{x} p^{x} (q)^{n-x}

where,

- n = the number of experiments
- x = 0, 1, 2, 3, 4, …
- p = Probability of success in a single experiment
- q = Probability of failure in a single experiment (= 1 – p)

The binomial distribution formula is also written in the form of n-Bernoulli trials, where ^{n}C_{x} = n!/x!(n-x)!. Hence, P(x:n,p) = n!/[x!(n-x)!].p^{x}.(q)^{n-x}

## Examples on Binomial Distribution Formula

**Example 1: If a coin is tossed 5 times, using binomial distribution find the probability of:**

**(a) Exactly 2 heads**

**(b) At least 4 heads.**

**Solution:**

**(a) **The repeated tossing of the coin is an example of a Bernoulli trial. According to the problem:

Number of trials: n=5

Probability of head: p= 1/2 and hence the probability of tail, q =1/2

For exactly two heads:

x=2

P(x=2) = ^{5}C2 p^{2} q^{5-2 }= 5! / 2! 3! × (½)^{2}× (½)^{3}

P(x=2) = 5/16

**(b) **For at least four heads,

x ≥ 4, P(x ≥ 4) = P(x = 4) + P(x=5)

Hence,

P(x = 4) = ^{5}C4 p^{4} q^{5-4} = 5!/4! 1! × (½)^{4}× (½)^{1} = 5/32

P(x = 5) = ^{5}C5 p^{5} q^{5-5} = (½)^{5} = 1/32

**Answer: Therefore, P(x ≥ 4) = 5/32 + 1/32 = 6/32 = 3/16**

**Example 2: For the same question given above, find the probability of getting at most 2 heads.**

**Solution:**

Solution: P(at most 2 heads) = P(X ≤ 2) = P (X = 0) + P (X = 1)

P(X = 0) = (½)^{5} = 1/32

P(X=1) = 5C1 (½)^{5}.= 5/32

**Answer:** **Therefore, P(X ≤ 2) = 1/32 + 5/32 = 3/16**

**Example 3: 60% of people who purchase sports cars are men. Find the probability that exactly 7 are men if 10 sports car owners are randomly selected.**

**Solution:**

Let’s Identify ‘n’ and ‘X’ from the problem.

The number of sports car owners are randomly selected is n = 10, and

The number to find the probability is X = 7.

Given: p = 60%, or 0.6.

Therefore, the probability of failure is q = 1 – 0.6 = 0.4

Now, using the binomial distribution formula

\( P( x ) = \frac{{n!}}{{( {n - x} )!x!}}.( p )^x .( q)^{n - x} \\

= \frac{{10!}}{{( {10 - 7} )!7!}}.( {0.6} )^7 .t( {0.4})^{10 - 7} \\

= 120 \times 0.0279936 \times 0.064 \\

= 0.215 \)

**Answer: The probability that exactly 7 are men is 0.215 or 21.5%.**

## FAQs on Binomial Distribution Formula

### What Is Binomial Distribution and Binomial Distribution Formula in Statistics?

The binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as the normal distribution. The binomial distribution, therefore, represents the probability for x successes in n trials, given a success probability p for each trial. The binomial distribution formula is for any random variable X, given by; P(x:n,p) = ^{n}C\(_x\) p^{x }(1-p)^{n-x} **Or** P(x:n,p) = ^{n}C\(_x\) p^{x} (q)^{n-x}, where, n is the number of experiments, p is probability of success in a single experiment, q is probability of failure in a single experiment (= 1 – p) and takes values as 0, 1, 2, 3, 4, …, n.

### What Is the Purpose of the Binomial Distribution Formula?

The binomial distribution formula allows us to compute the probability of observing a specified number of "successes" when the process is repeated a specific number of times (e.g., in a set of patients) and the outcome for a given patient is either a success or a failure.

### What Is the Formula for Binomial Distribution?

The formula for binomial distribution is:

P(x: n,p) = ^{n}C\(_x\) p^{x} (q)^{n-x}

Where p is the probability of success, q is the probability of failure, n = number of trials.

### What Is the Binomial Distribution Formula for the Mean and Variance?

The mean and variance of the binomial distribution are:

Mean = np

Variance = npq

where p is the probability of success, q is the probability of failure, n = number of trials.

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