Compreendendo a Distribuição Binomial
What Is the Binomial Distribution?
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It is one of the most fundamental discrete probability distributions. A classic example is flipping a coin n times and counting how many heads appear. The distribution is defined by two parameters: n (number of trials) and p (probability of success on each trial).
The Binomial Formula
The probability of exactly k successes in n trials is given by P(X=k) = C(n,k) × p^k × (1-p)^(n-k), where C(n,k) is the binomial coefficient (combination). This formula multiplies the number of ways to arrange k successes among n trials by the probability of any specific arrangement. The sum of all probabilities from k=0 to k=n equals exactly 1.
Conditions for Binomial Distribution
Four conditions must be met: fixed number of trials (n), each trial has only two possible outcomes (success/failure), the probability of success (p) is constant across all trials, and trials are independent (one outcome does not affect another). When these conditions are met, the binomial distribution accurately models the situation.
Mean, Variance, and Standard Deviation
The mean of a binomial distribution is μ = np, representing the expected number of successes. The variance is σ² = np(1-p), and the standard deviation is σ = √(np(1-p)). For example, with n=100 and p=0.5, the mean is 50 and the standard deviation is 5. These parameters help predict the range of likely outcomes and determine confidence intervals.
Normal Approximation
When n is large and p is not too close to 0 or 1, the binomial distribution approximates a normal distribution with mean np and standard deviation √(np(1-p)). This approximation is generally acceptable when np ≥ 5 and n(1-p) ≥ 5. For small n or extreme p values, the exact binomial calculation is necessary for accuracy.