Final answer:
The binomial probability formula calculates the probability of getting a specific number of successes in a series of independent trials, factoring in the success and failure rates for each trial.
Step-by-step explanation:
Binomial Probability Formula
The binomial probability formula is used to find the probability of exactly x successes in n independent trials, with the probability of success being p on each trial. The two sides of this formula provide a way to calculate the likelihood of a particular outcome in a binomial experiment. The formula is expressed as:
P(X = x) = (n choose x)pxqn-x
Where:
- X is the number of successes
- n is the total number of trials
- p is the probability of success on a single trial
- q is the probability of failure on a single trial (q = 1 - p)
- (n choose x) is a binomial coefficient calculated as n!/(x!(n-x)!)
The mean (µ) of the distribution is calculated using µ = np, and the standard deviation (σ) is calculated using σ = √npq.
The formula signifies that the probability of getting exactly x successes in n trials of a binomial experiment is equal to the number of ways to choose x successes, multiplied by the probability of success raised to the power of x, multiplied by the probability of failure raised to the power of n-x.