Binomial Distribution

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The binomial distribution is a probability distribution that describes the likelihood of a given number of successes in a fixed number of trials. For example, in the case of flipping fair coins, the binomial distribution can be used to calculate the probability of getting a certain number of heads in a certain number of coin flips.

Properties of Binomial Distribution:

The binomial distribution is a discrete probability distribution that is defined by two parameters: the number of trials and the probability of success on each trial. It has several important properties, including:

  • Discrete values: The binomial distribution only takes on integer values. This means that the number of successes in a given number of trials can only be an integer, such as 0, 1, 2, etc.
  • Two possible outcomes: The binomial distribution assumes that each trial has only two possible outcomes: success or failure. The probability of success on each trial is constant across all trials.
  • Fixed number of trials: The binomial distribution assumes a fixed number of independent trials.
  • Independence of trials: The trials are independent; the outcome of one trial does not affect the outcome of other trials. This means that the probability of success is the same for each trial.
  • Probability of a given number of successes: The binomial distribution allows you to calculate the probability of a given number of successes in a given number of trials. For example, you can use the binomial distribution to calculate the probability of getting 5 heads in 10 coin flips.
  • Mean and variance: The binomial distribution has a mean equal to the number of trials multiplied by the probability of success on each trial (n.p) and a variance equal to the number of trials multiplied by the probability of success on each trial multiplied by the probability of failure on each trial or n.p.(1-p).

These are some of the most important properties of binomial distribution.

Probability Mass Function (PMF) - Binomial Distribution

This is the formula for the probability mass function of a binomial distribution. It gives the probability of x successes in n trials, where the probability of success on each trial is p. The formula is derived from the probability of success and failure on each trial, and it considers that the outcomes of each trial are independent.

$$P(x) = \frac{n!}{x! \cdot (n - x)!} \cdot p^x \cdot (1 - p)^{(n - x)}$$

where:

  • \(P(x)\) is the probability of x successes in n trials
  • \(n\) is the total number of trials
  • \(x\) is the number of successes
  • \(p\) is the probability of success on each trial
  • \(!\) is the factorial symbol, which denotes the product of all positive integers less than or equal to the number. For example, \(4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24\).

The formula uses factorials (represented by the exclamation mark) to calculate the number of possible combinations of successes and failures in the experiment. The formula is often simplified to make it easier to work with, but the basic idea is to calculate the probability of a given number of successes in the experiment.

The formula for the binomial distribution can be rewritten using combinations (represented by the notation nCx) to make it easier to understand and work with. The rewritten formula is as follows:

Binomial Distribution Formula with Combinations:

$$P(x) = nCx \cdot p^x \cdot (1 - p)^{(n - x)}$$

where:

  • \(P(x)\) is the probability of x successes in n trials
  • \(n\) is the total number of trials
  • \(x\) is the number of successes
  • \(p\) is the probability of success on each trial
  • \(nCx\) is the binomial coefficient, which is defined as \(nCx = \frac{n!}{x! \cdot (n - x)!}\). It gives the number of ways to choose x successes from n trials.

Mean, Standard Deviation and Variance of Binomial Distribution

The mean, variance, and standard deviation are important statistical measures that describe the characteristics of a probability distribution. In the case of the binomial distribution, the mean, variance, and standard deviation can be calculated using the following formulas:

Mean of a Binomial Distribution:

Mean = \(n * p\)

Where:

n is the number of trials

p is the probability of success in each trial

Standard Deviation of a Binomial Distribution:

Standard Deviation = \(\sqrt{n * p * (1 - p)}\)

Where:

n is the number of trials

p is the probability of success in each trial

(1 - p) is the probability of failure in each trial

Variance of a Binomial Distribution:

Variance = \(n * p * (1 - p)\)

Where:

n is the number of trials

p is the probability of success in each trial

(1 - p) is the probability of failure in each trial

 

Using Microsoft Excel

To calculate the probability of a binomial event using Microsoft Excel, you can use the BINOM.DIST function, which calculates the probability of a specified number of successes in a fixed number of Bernoulli trials.

BINOM.DIST

To use the BINOM.DIST(number_s, trials, probability_s, cumulative) function, you need to provide the following input arguments:

  • number_s: The number of successes you want to calculate the probability for.
  • trials: The total number of Bernoulli trials.
  • probability_s: The probability of success on each trial.
  • cumulative: A logical value that specifies whether to return the probability of the specified number of successes (FALSE) or the probability of the specified number of successes or fewer (TRUE).

For example, to calculate the probability of getting exactly 3 heads in 5 coin flips with a probability of heads on each flip of 0.5, you would use the following formula:

=BINOM.DIST(3, 5, 0.5, FALSE) = 0.3125

This formula uses the BINOM.DIST function to calculate the probability of getting exactly 3 heads in 5 flips, with a probability of heads on each flip of 0.5.

If you want to determine the probability of 3 or fewer heads in 5 coin flips, you will use cumulative as TRUE.

=BINOM.DIST(3, 5, 0.5, TRUE) = 0.8125


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