Final answer:
A random variable Y is not binomial when drawing cards without replacement from a deck because the trials aren't independent and the probability of success changes. However, the variable X could be binomial for daily plays in a pick 3 game over a year if each play has independent outcomes and the probability of winning doesn't change each day.
Step-by-step explanation:
To determine whether the random variable Y is a binomial random variable, we can review the criteria for a binomial distribution. The criteria include having a fixed number of independent trials, only two outcomes (success or failure), and the probability of success remains constant in each trial.
a) Since 1 card is drawn without replacement from a 52 card deck and X is the number of aces, the trials are not independent (removing a card affects the next draw), and the probability of success changes after each draw. Therefore, Y is not a binomial random variable for this scenario.
b) If Joe plays pick 3 every day for a year, assuming each play is independent of the others (the outcome of one does not affect the others) and the probability of winning remains the same each day, the random variable X would be the number of times he wins in these independent trials. The values of n and p would be the total number of days he played in the year and the probability of winning on any given day, respectively. Here, X could be a binomial random variable.