Two cards are selected, one at a time from a standard deck of 52 cards. Let x represent the number of Jacks drawn in a set of 2 cards.
(A) If this experiment is completed without replacement, explain why x is not a binomial random variable.
(B) If this experiment is completed with replacement, explain why x is a binomial random variable
(A) If the experiment is completed without replacement, x is not a binomial random variable because in a binomial experiment, each trial must be independent with equal probability of success. In this case, the probability of drawing a Jack changes after the first card is drawn. For example, if the first card drawn is a Jack, the probability of drawing another Jack is reduced because there is one less Jack in the deck. Therefore, the probability of success (drawing a Jack) varies depending on the outcome of the first draw, and thus violates the condition of equal probability of success for each trial.
(B) If the experiment is completed with replacement, x is a binomial random variable. In this case, after each card is drawn, it is placed back into the deck, and the deck is shuffled before drawing the next card. When we shuffle the deck, each card has an equal probability of being drawn again. Therefore, the probability of drawing a Jack remains constant for each trial, independent of the previous draws. Since each trial (drawing a card) is independent and has the same probability of success (drawing a Jack), the number of Jacks drawn in a set of 2 cards follows a binomial distribution.