Binomial Distribution
Binomial Distribution
The binomial distribution is a probability distribution that describes the number of successes in a sequence of n independent trials, each of which has a probability of success p.
Assumptions:
- The trials are independent.
- Each trial has a constant probability of success p.
- The number of trials n is finite.
Formula:
The probability of getting exactly k successes in n trials is given by the following formula:
P(X = k) = n choose k * p^k * (1-p)^(n-k)
where:
- P(X = k) is the probability of getting exactly k successes
- n is the number of trials
- k is the number of successes
- p is the probability of success
- n choose k is the binomial coefficient, given by the formula:
n choose k = n! / (k! * (n-k)! )
Moments:
The mean (expected value) of a binomial distribution is given by:
E = np
The variance of a binomial distribution is given by:
Var = np(1-p)
Applications:
The binomial distribution is used to model a wide variety of phenomena, including:
- The number of defective items in a batch of products
- The number of heads in a coin toss
- The number of successes in a raffle
Example:
Suppose you have a coin with a probability of heads of 0.4 and you toss it 10 times. The number of heads that you will get in 10 tosses will follow a binomial distribution with n = 10 and p = 0.4. The probability of getting exactly 3 heads is given by:
P(X = 3) = 10 choose 3 * 0.4^3 * (1-0.4)^(10-3) = 0.216
This means that there is a 21.6% chance of getting exactly 3 heads in 10 tosses of a coin with a probability of heads of 0.4.
