Binomial distribution
Binomial Distribution
The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. A Bernoulli trial is an experiment that results in a binary outcome: success or failure.
Characteristics of a Binomial Experiment
A binomial experiment has the following characteristics:
- The number of trials, ( n ), is fixed.
- Each trial has only two possible outcomes: success or failure.
- The probability of success, ( p ), is the same for each trial.
- The trials are independent; the outcome of one trial does not affect the outcome of other trials.
Binomial Probability Formula
The probability of getting exactly ( k ) successes in ( n ) trials is given by the binomial probability formula:
[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} ]
where:
- ( P(X = k) ) is the probability of ( k ) successes in ( n ) trials.
- ( \binom{n}{k} ) is the binomial coefficient, which can be calculated as ( \frac{n!}{k!(n-k)!} ).
- ( p ) is the probability of success on a single trial.
- ( 1-p ) is the probability of failure on a single trial.
- ( n ) is the number of trials.
- ( k ) is the number of successes ( ( k = 0, 1, 2, \ldots, n ) ).
Mean and Variance of Binomial Distribution
The mean (( \mu )) and variance (( \sigma^2 )) of a binomial distribution can be calculated using the following formulas:
- Mean: ( \mu = np )
- Variance: ( \sigma^2 = np(1-p) )
Binomial Distribution vs. Other Distributions
Here's a table comparing the binomial distribution with other common distributions:
| Feature/Distribution | Binomial Distribution | Poisson Distribution | Normal Distribution |
|---|---|---|---|
| Type | Discrete | Discrete | Continuous |
| Number of Outcomes | Two (success/failure) | Potentially infinite | Infinite |
| Trials | Fixed number | Infinite | Not applicable |
| Probability of Success | Same for each trial | Varies with interval | Not applicable |
| Mean | ( np ) | ( \lambda ) | ( \mu ) |
| Variance | ( np(1-p) ) | ( \lambda ) | ( \sigma^2 ) |
Example of Binomial Distribution
Let's consider a simple example to illustrate the binomial distribution.
Example: Suppose you flip a fair coin 5 times. What is the probability of getting exactly 3 heads?
Here, we have a binomial experiment where:
- ( n = 5 ) (number of trials)
- ( k = 3 ) (number of successes)
- ( p = 0.5 ) (probability of success on a single trial)
Using the binomial probability formula:
[ P(X = 3) = \binom{5}{3} (0.5)^3 (1-0.5)^{5-3} ]
[ P(X = 3) = \frac{5!}{3!2!} (0.5)^3 (0.5)^2 ]
[ P(X = 3) = 10 \times (0.5)^5 ]
[ P(X = 3) = 10 \times 0.03125 ]
[ P(X = 3) = 0.3125 ]
So, the probability of getting exactly 3 heads in 5 coin flips is 0.3125 or 31.25%.
Conclusion
The binomial distribution is a fundamental probability distribution used to model situations where there are a fixed number of independent trials, each with two possible outcomes. Understanding the binomial distribution is essential for various applications in statistics, finance, and other fields where probability and risk assessment are crucial.