Binomial distribution

Definition

The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is used to model situations where there are two possible outcomes, such as success or failure, and provides a way to visualize the likelihood of different outcomes based on varying parameters like the number of trials and probability of success.

5 Must Know Facts For Your Next Test

1. The binomial distribution is defined by two parameters: n (the number of trials) and p (the probability of success on each trial).
2. The mean of a binomial distribution can be calculated using the formula $$\mu = n \times p$$.
3. The variance is given by $$\sigma^2 = n \times p \times (1 - p)$$, which indicates the spread of the distribution.
4. The shape of the binomial distribution can vary; it can be symmetric, skewed right, or skewed left depending on the values of n and p.
5. As n increases, the binomial distribution approaches a normal distribution when p is not too close to 0 or 1, due to the Central Limit Theorem.

Review Questions

• How do changes in the parameters n (number of trials) and p (probability of success) affect the shape and properties of the binomial distribution?
  • Changes in the parameters n and p significantly influence the shape of the binomial distribution. Increasing n while keeping p constant generally makes the distribution more symmetric and bell-shaped. Conversely, if p is very small or very large, increasing n tends to create a right or left skew, respectively. Additionally, adjusting p affects the peak's location; higher values shift it toward more successes while lower values do the opposite.
• Explain how to calculate the probability of obtaining exactly k successes in a binomial distribution.
  • To calculate the probability of obtaining exactly k successes in a binomial distribution, you use the formula $$P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}$$. Here, $$\binom{n}{k}$$ represents the binomial coefficient, which counts how many ways k successes can occur in n trials. This formula combines both the probability of success raised to k and the probability of failure raised to the remaining trials, ensuring all scenarios are considered.
• Discuss how the binomial distribution can be approximated using a normal distribution and under what conditions this approximation is valid.
  • The binomial distribution can be approximated by a normal distribution when certain conditions are met: specifically, when both np and n(1 - p) are greater than 5. This ensures that there are enough expected successes and failures for the normal approximation to hold. The approximation becomes useful for simplifying calculations and analyzing probabilities, especially when dealing with large sample sizes where direct computation would be cumbersome.

Related terms

Bernoulli Trial: A random experiment with exactly two possible outcomes: success or failure.

Probability Mass Function: A function that gives the probability that a discrete random variable is equal to a specific value.

Binomial Coefficient: A coefficient representing the number of ways to choose k successes in n trials, calculated as 'n choose k'.