Final answer:
A scenario can be represented by a binomial distribution if it involves a fixed number of trials with two possible outcomes per trial (success or failure) and if all trials are independent with constant probabilities of each outcome.
Step-by-step explanation:
To determine whether a scenario can be represented by a binomial distribution, it must fulfill certain criteria.
- There must be a fixed number of trials (n).
- There are only two possible outcomes for each trial, commonly labeled as success and failure.
- All trials are independent, meaning the outcome of one trial does not affect another, and the probability of success (p) and failure (q) remains constant across trials.
When these conditions are satisfied, the random variable X, representing the number of successes in n trials, has a binomial distribution with mean μ = np and standard deviation σ = √npq. The probability of exactly x successes is given by P(X = x).
For example, flipping a coin gives two outcomes, head or tail, with equal probability, and each flip is independent. Randomly guessing an answer on a multiple-choice question with four options is also a binomial situation if you define success as guessing correctly. In contrast, a situation where the number of outcomes is more than two or the trials are not independent would not meet the criteria for binomial distribution.