Binomial Distribution Introduction and Formula

Binomial Distribution

Binomial Distribution is a chapter under probability and statistics, which is tested in the A-level syllabus. Under Section B, 3.1 of the syllabus, students are taught content which Include:

 

  • Knowledge of the binomial expansion of (a + b)n for positive integer n
  • Binomial random variable as an example of a discrete random variable
  • Concept of binomial distribution B(n, p) and useof B(n, p) as a probability model, including conditions under which the binomial distribution is a suitable model
  • Use of mean and variance of a binomial distribution (without proof) 

What is a Binomial Distribution?

A binomial distribution is a frequency distribution of the possible number of successful outcomes in a given number of trials in each of which there is the same probability of success.

All binomial model should follow the following properties: 

  • The model consists of n repeated trials
  • Each trial has only 2 possible mutually exclusive outcomes, success or failure. 
  • The probability of success, denoted by p is the same in each trial, 
  • The repeated trials are independent.

Formula for Binomial Distribution is indicated as X~ B(n,p)

Where 

p is the probability of success in each individual trial.

q is the probability of failure in each individual trial, which is (1 – p).

k is the number of successful outcomes you want to calculate the probability for.

To use normal distribution to approximate the Binomial Distribution:

  1. Check that n is sufficiently large 
  2. Change the form of X~B (n,p) to become X~N (np, np(1-p)) approx. 
  3. Express the question in terms of P (X = r) or P ( X < r) or p(X > r) or P (X ≥ r) or P(a ≤ X ≤ b). 
  4. Use GC to work out the answer, which should be expressed with the continuity correction (+- 0.5). 

Binomial Distribution Example 

A factory produces light bulbs, and it is known that 5% of the bulbs are defective. If a sample of 10 bulbs is randomly selected, what is the probability that exactly 2 of them will be defective?

Solution:

To solve this problem using the binomial distribution formula, we need to identify the parameters:

n = 10 (number of trials, i.e., the size of the sample)

p = 0.05 (probability of success, i.e., the probability that a bulb is defective)

k = 2 (number of successes we want to find, i.e., exactly 2 defective bulbs)

The probability of getting exactly 2 defective bulbs (P(X = 2)) can be calculated using the binomial probability mass function:

P(X = k) = C(n, k) * p^k * q^(n-k)

where q = 1 – p.

Plugging in the values:

P(X = 2) = C(10, 2) * (0.05)^2 * (1 – 0.05)^(10 – 2)

Now, calculate the binomial coefficient C(10, 2):

C(10, 2) = 10! / (2! * (10 – 2)!) = 10! / (2! * 8!) = (10 * 9) / 2 = 45

Now, calculate the probability:

P(X = 2) = 45 * (0.05)^2 * (0.95)^8 ≈ 0.0746

So, the probability that exactly 2 out of 10 randomly selected light bulbs will be defective is approximately 0.0746, or 7.46%.

Sample Question for Binomial Distribution (JC Prelims)

ACJC Prelims 2009

In a current Affair’s Quiz for 200 JC2 students, there are 20 questions. Each question is followed by five possible answers, of which only one is correct. The students have to decide for each question, which one of the five answers is correct. Assuming that all the students answered the questions randomly. 

  1. i) Find the probability that a randomly selected student has at least 15 incorrect answers;
  2. ii) Find, using a suitable approximation, the probability that the total number of correct answers given by all JC2 students is at most 820. 

Answers:
(i): P(X ≥ 15) = 0.804

(ii) P(X≤820) = 0.791

How is Binomial Distribution Used in Real Life?

Quality Control: In manufacturing and production industries, quality control is crucial to ensure that products meet certain standards. The binomial distribution helps in analyzing the number of defective items in a sample from a production batch, allowing companies to estimate the quality of their products and make necessary improvements.

Market Research: Binomial distribution can be employed in market research to understand customer preferences and behaviors. For instance, it can be used to estimate the probability of a specific target audience responding positively to a product or advertisement.

A/B Testing in Digital Marketing: Binomial distribution is widely used in A/B testing, where two variants of a web page, advertisement, or email are compared to determine which one performs better in terms of user engagement or conversion rates.

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