Which of the following situations describes a binomial random variable?

Question

BlackTom AI · Accepted Answer

Question restatement: Which of the following situations describes a binomial random variable?

Option A: The Youth Risk Behavior Surveillance System Survey from 2013 estimates that 19.4% of teenagers reported to have experienced bullying. Assume that whether or not one teenager was bullied is independent of whether or not another teenager was bullied. You count the number of times a randomly selected teenager was bullied on average in 2013.
- Why this is not binomial: While there is a fixed probability of bullying for each teenager, the scenario describes counting the number of times a randomly selected teenager was bullied, which is essentially a single Bernoulli trial outcome, not a fixed number of independent trials. The “average” across teenagers suggests a mean rather than a binomial count over a fixed number of trials, so it does not neatly fit the binomial framework of n identical, independent trials with a fixed number of successes.

Option B: A new restaurant opening in Emory Village has a 30% chance of survival during their first year. Assume one restaurant surviving is independent to another one surviving. You count the number of restaurants that stay open for multiple years.
- Why this is not binomial: Although each restaurant has a fixed probability of survival, the event being counted—staying open for multiple years—could involve multiple time horizons and is not clearly a single fixed trial with identical outcomes. Moreover, the phrase "multiple years" introduces a different time structure than a single Bernoulli trial per restaurant, so the count over restaurants does not necessarily constitute a standard binomial scenario as stated.

Option C: You play two games against the same opponent. The probability that you win the first game is 0.4. If you win the first game, the probability you also win the second game is 0.2. You are interested in the probability that you win both games.
- Why this is not binomial: In a binomial model, trials are independent and have the same probability of success. Here, the probability of winning the second game depends on the outcome of the first (P(win2 | win1) = 0.2), so the trials are not independent and do not share a common p. Therefore, it fails the core binomial assumptions.

Option D: You are training your dog to catch a ball. The dog either catches the ball or fails to catch the ball. Assume that catching the ball in each attempt is independent. You count the number of times your dog catches the ball out of 6 throws.
- Why this is binomial (the correct framing): Here we have a fixed number of independent trials (n = 6 throws), each with the same probability of success (catching the ball) on every throw. The variable being counted is the number of successes (catches) among the 6 trials, which is the quintessential binomial random variable with parameters n = 6 and p equal to the per-throw probability of catching the ball. This setup matches the binomial distribution exactly: fixed n, independent trials, and identical success probability across trials.

Summary observations: Options A, B, and C each violate one or more key binomial assumptions (fixed number of independent trials with identical success probability). Option D satisfies all the binomial criteria, making it the correct description of a binomial random variable.

FA25_QTM_100_3 Final Exam- Requires Respondus LockDown Browser

Which of the following situations describes a binomial random variable?

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