Three cards are selected, one at a time from a standard deck of 52 cards. Let x represent the number of tens drawn in a set of 3 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) In this experiment, three cards are selected without replacement. This means that after each card is drawn, it is not returned to the deck. Therefore, the probability of drawing a ten changes after each draw.
For the first card, the probability of drawing a ten is 4/52, since there are 4 tens out of 52 cards. After the first ten is drawn, the deck only has 51 cards, with 3 tens remaining. So, for the second card, the probability of drawing a ten is 3/51. Finally, after the second ten is drawn, the deck has 50 cards, with 2 tens remaining. Therefore, for the third card, the probability of drawing a ten is 2/50.
Since the probability of drawing a ten is changing after each draw, x is not a binomial random variable, which requires a fixed probability of success for each trial.
(B) If the experiment is completed with replacement, it means that after each card is drawn, it is returned to the deck before the next draw. This ensures that the probability of drawing a ten remains the same for each card.
In this case, the probability of drawing a ten is always 4/52, regardless of the previous draws. The draws are independent of each other, and the probability of success (drawing a ten) remains constant for each trial.
Since the experiment satisfies the requirements for a binomial random variable - fixed probability of success, independent trials, and a fixed number of trials - x becomes a binomial random variable in this situation.