The binomial distribution is a fundamental probability distribution used in statistics and probabilistic studies. It models the number of successes in a fixed number of independent Bernoulli trials, which are experiments that have exactly two outcomes: success and failure. In our example, a sample size of 50 was taken from a binomial distribution where the number of trials, denoted by \( n \), is 5.
- The number of trials (\( n \)) is the number of times an experiment is performed, which is 5 for each observation in the given exercise.
- Each trial has exactly two possible outcomes: success ("success" here is achieving a count of outcomes with \( x \geq 3\)) or failure ("failure" would be outcomes where \( x < 3 \)).
- The probability of success (\( p \)) remains constant for each trial.
When working with the binomial distribution, we often use it to determine the probability of obtaining a certain number of successes across multiple trials. In this problem, we're particularly interested in calculating the probability for cases where the number of successes is 3, 4, or 5.