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) If the experiment is completed without replacement, the outcome of each draw is dependent on the previous draws. In a binomial random variable, each trial must be independent, meaning that the outcome of one trial does not affect the outcome of the others. In this case, the probability of drawing a ten changes after each draw since there are fewer cards left in the deck.
(B) If the experiment is completed with replacement, it means that after each draw, the card is placed back into the deck, and the number of cards remains constant. Each draw becomes independent and has the same probability of success, which is the probability of drawing a ten. Therefore, in this scenario, x can be considered as a binomial random variable since each trial is independent and has the same probability of success.
A) If the experiment is completed without replacement, x is not a binomial random variable because the probability of drawing a ten on each trial changes after each card is drawn. In a binomial distribution, each trial must be independent and have the same probability of success. In this case, after each card is drawn, the number of tens remaining in the deck decreases, which changes the probability of drawing a ten on the next trial.
To calculate the probability of drawing a ten in each trial without replacement, you would need to use the concept of conditional probability. This means that the probability of drawing a ten in the first trial would depend on whether or not a ten was drawn in the previous trial, making it different from a binomial random variable.
B) If the experiment is completed with replacement, x would be a binomial random variable. In this case, after each card is drawn, it is returned to the deck before the next trial, ensuring that the probability of drawing a ten remains the same for each trial.
To determine the probability of drawing a ten in each trial with replacement, you can simply consider each trial as an independent event with a fixed probability of success (i.e., drawing a ten). The number of tens drawn in these independent trials would follow a binomial distribution, where each trial has the same probability of success (the probability of drawing a ten in a single trial) and the number of trials (the number of cards drawn) is fixed.