1. Meaning of a Binomial Distribution
A binomial distribution describes the probability of getting a certain number of successes in a fixed number of repeated trials. Each trial has only two possible outcomes — usually called success and failure.
It is useful when the same action is repeated again and again under identical conditions.
1.1. Example
If a coin is tossed 5 times, each toss can give Head or Tail. The number of Heads (0 to 5) follows a binomial distribution.
2. Conditions for a Binomial Distribution
A random variable follows a binomial distribution only when these conditions hold:
2.1. Fixed Number of Trials
The experiment is repeated a fixed number of times, say \( n \).
Example: tossing a coin 10 times.
2.2. Each Trial Has Two Outcomes
Every trial results in either success or failure.
Example: Head or Tail, defective or non-defective.
2.3. Constant Probability of Success
The probability of success \( p \) remains the same in every trial.
Example: probability of getting a head is always 1/2.
2.4. Trials Are Independent
The outcome of one trial does not affect another.
Example: one toss of a coin does not influence the next toss.
3. The Binomial Probability Formula
If X is the number of successes in n trials, then the binomial probability of getting exactly k successes is:
\( P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \)
Here:
- \( n \) = number of trials
- \( k \) = number of successes
- \( p \) = probability of success
- \( 1 - p \) = probability of failure
- \( \binom{n}{k} \) = number of ways to choose k successes
3.1. Example
A coin is tossed 3 times. What is the probability of getting exactly 2 heads?
Here:
- n = 3
- k = 2
- p = 1/2
\( P(X = 2) = \binom{3}{2}(1/2)^2 (1/2)^1 = 3/8 \)
4. Mean and Variance of a Binomial Distribution
The binomial distribution has simple formulas for its average (mean) and spread (variance):
\( \text{Mean } (\mu) = np \)
\( \text{Variance } (\sigma^2) = np(1 - p) \)
4.1. Example
If n = 10 and p = 0.3, then:
- Mean = \( 10 \times 0.3 = 3 \)
- Variance = \( 10 \times 0.3 \times 0.7 = 2.1 \)
5. More Examples
Some situations where the binomial distribution applies:
5.1. Repeated Coin Tosses
Number of heads in 20 tosses follows a binomial distribution with n = 20, p = 1/2.
5.2. Defective Items in a Batch
If each item has a fixed probability of being defective, the number of defective items in a sample of n items follows a binomial distribution.
5.3. Correct Answers in Guessing
If each question has two options and guesses are random, the number of correct answers in n questions is binomial.