Final answer:
Scenarios a) and c) represent binomial random variables due to their fixed number of independent trials, two possible outcomes, and constant probability of success. Scenario b) is not a binomial random variable because the probability of winning varies between games.
Step-by-step explanation:
To determine whether each scenario represents a binomial random variable, we must ascertain if there are a fixed number of independent trials with only two possible outcomes (success or failure) and if the probability of success remains constant for all trials.
- a) Rolling a die 25 times and counting the number of times a "3" is rolled can be seen as a binomial random variable. Here, we have a fixed number of trials (25 rolls), each independent from the other, with two possible outcomes (rolling a 3 or not rolling a 3), and the probability of rolling a 3 is constant (1/6) each time we roll the die.
- b) The number of games the SJSU football team wins is not a binomial random variable because the probability of success (winning a game) changes depending on whether the game is a conference or a nonconference game (70% chance versus 60% chance). A binomial distribution requires that the probability of success be the same for each trial.
- c) Evelyn getting an A in each class she takes over four years can be considered as a binomial random variable, assuming she takes the same number of classes every term, and the probability of getting an A (75%) does not change for each class throughout the four years.
Thus, scenarios a) and c) represent binomial random variables, while b) do not meet the criteria for a binomial random variable because the probability of success is not constant.