Binomial Distribution Formula: Probability, Negative, Examples

In probability theory and statistics, the binomial distribution is the discrete probability distribution that gives only two possible results in an experiment, either Success or Failure . For example, if we toss a coin, there could be only two possible outcomes: heads or tails, and if any test is taken, then there could be only two results: pass or fail. This distribution is also called a binomial probability distribution.

There are two parameters n and p used here in a binomial distribution. The variable ‘n’ states the number of times the experiment runs and the variable ‘p’ tells the probability of any one outcome. Suppose a die is thrown randomly 10 times, then the probability of getting 2 for anyone throw is ⅙. When you throw the dice 10 times, you have a binomial distribution of n = 10 and p = ⅙. Learn the formula to calculate the two outcome distribution among multiple experiments along with solved examples here in this article.

Binomial Probability Distribution

In the binomial probability distribution, we examine the number of 'Successes' in a series of n experiments. Each experiment involves posing a yes-no question, and the binary outcome is represented as either a success/yes/true/one (with probability p) or a failure/no/false/zero (with probability q = 1 − p). A single trial of success or failure is also known as a Bernoulli trial or Bernoulli experiment, and when multiple such trials are conducted sequentially, it forms a Bernoulli process. When n = 1, meaning a single experiment, the binomial distribution is essentially a Bernoulli distribution. The binomial distribution serves as the foundation for the well-known binomial test used for assessing statistical significance.

Negative Binomial Distribution

In probability theory and statistics, the negative binomial distribution pertains to the number of successes achieved in a sequence of independent and identically distributed Bernoulli trials before a specific count of failures occurs. In this context, the number of failures is denoted as 'r.' For instance, consider the scenario of rolling a die, where the occurrence of a 1 is regarded as a failure, and all other outcomes (non-1's) are considered successes. If we repeatedly roll the die until we observe the appearance of a 1 for the third time (i.e., r = three failures), then the probability distribution governing the count of non-1 outcomes that occurred would be described by the negative binomial distribution.

Binomial Distribution Examples

As previously discussed, the binomial distribution provides a framework for assessing the likelihood of various outcomes. In practical applications, this concept is frequently employed for:

Determining the quantity of raw and utilized materials required in the production of a product.

Conducting surveys to gather positive and negative feedback from the public about a specific product or location.

Utilizing YES/NO surveys to gauge the number of individuals who favor a particular television channel.

Estimating the count of male and female employees within an organization.

Calculating the number of votes received by a candidate in an election, based on binary outcomes (0 or 1 probability).

Binomial Distribution Formula

The binomial distribution formula expresses the probability of a random variable X taking on a specific value x, given by:

P(x;n,p) = nCx * p^x * (1-p)^(n-x)

or equivalently,

P(x;n,p) = nCx * p^x * q^(n-x)

Here are the key variables involved:

n = the number of experiments or trials.

x = the specific number of successes you want to calculate the probability for, ranging from 0 to n.

p = the probability of success in a single experiment or trial.

q = the probability of failure in a single experiment, calculated as q = 1 - p.

The binomial distribution formula can also be expressed in terms of n-Bernoulli trials, where nCx represents the number of combinations and is calculated as n! / [x!(n-x)!]. So, you can write it as:

P(x;n,p) = (n! / [x!(n-x)!]) * p^x * q^(n-x)

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Binomial Distribution Mean and Variance

In the context of a binomial distribution, you can determine the mean, variance, and standard deviation for a given number of successes using the following formulas:

Mean (μ): μ = np, where:

n is the number of experiments or trials,

p is the probability of success.

Variance (σ²): σ² = npq, where:

n is the number of experiments or trials,

p is the probability of success,

q is the probability of failure, and q = 1 - p.

Standard Deviation (σ): σ = √(npq), where:

n is the number of experiments or trials,

p is the probability of success,

q is the probability of failure, and q = 1 - p.

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Binomial Distribution Vs Normal Distribution

The primary distinction between the binomial distribution and the normal distribution lies in their nature: the binomial distribution is discrete, while the normal distribution is continuous. This means that the binomial distribution deals with a finite number of discrete events, whereas the normal distribution encompasses an infinite number of continuous events. However, it's worth noting that when the sample size for the binomial distribution becomes very large, the distribution curve for the binomial distribution closely approximates the shape of the normal distribution curve.

Key Properties of the Binomial Distribution:

• Two Possible Outcomes: The binomial distribution is characterized by two distinct outcomes, such as true or false, success or failure, or yes or no.
• Fixed Number of Trials: It involves a fixed number of independent trials, denoted as 'n', or a specific number of trials conducted 'n' times.
• Constant Probability: The probability of success or failure remains consistent across all trials.
• Counting Successes: The binomial distribution focuses on counting the number of successes achieved out of 'n' independent trials.
• Independence of Trials: Each trial is independent, meaning the outcome of one trial does not influence the outcome of another trial.

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Binomial Distribution Examples And Solutions

Example 1:

(a) When a coin is tossed 5 times, we can apply the binomial distribution to find the probability of getting exactly 2 heads:

Number of trials: n = 5

Probability of getting a head (success): p = 1/2

Probability of getting a tail (failure): q = 1/2

For exactly 2 heads:

x = 2

P(x = 2) = 5C2 * (1/2)^2 * (1/2)^(5-2)

P(x = 2) = 10 * (1/4) * (1/8)

P(x = 2) = 5/16

(b) To find the probability of getting at least 4 heads:

For at least 4 heads, we need to calculate the probabilities of getting 4, 5, or 5 heads and sum them up.

P(x = 4) = 5C4 * (1/2)^4 * (1/2)^(5-4) = 5/32

P(x = 5) = 5C5 * (1/2)^5 * (1/2)^(5-5) = 1/32

P(x ≥ 4) = P(x = 4) + P(x = 5) = 5/32 + 1/32 = 6/32 = 3/16

Example 2:

To find the probability of getting at most 2 heads (0, 1, or 2 heads):

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

P(X = 0) = (1/2)^0 * (1/2)^10 = 1/1024

P(X = 1) = 10C1 * (1/2)^1 * (1/2)^9 = 10/1024

P(X = 2) = 10C2 * (1/2)^2 * (1/2)^8 = 45/1024

P(X ≤ 2) = 1/1024 + 10/1024 + 45/1024 = 56/1024 = 7/128

Example 3:

For a fair coin tossed 10 times:

Number of trials: n = 10

Probability of getting a head (success): p = 1/2

Probability of getting a tail (failure): q = 1/2

(i) Probability of getting exactly 6 heads:

P(X = 6) = 10C6 * (1/2)^6 * (1/2)^(10-6) = 210/1024 = 105/512

(ii) Probability of getting at least 6 heads:

P(X ≥ 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)

P(X ≥ 6) = (105/512) + (45/512) + (10/512) + (1/512) + (1/512) = 162/512 = 81/256