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, x is not a binomial random variable.
To understand why, let's first define what a binomial random variable is. A binomial random variable is a discrete random variable that represents the number of successes in a fixed number of independent Bernoulli trials, where each trial has two possible outcomes - success or failure - with the same probability of success.
In the given scenario, the first card selected has an equal chance of being a ten or non-ten. However, for the second and third cards, the probabilities change based on the outcome of the previous draws. Since the cards are not replaced after each draw, the sample space decreases with each card drawn, and the probabilities are affected by the previous outcomes. Therefore, the trials are not independent, which violates the condition for a binomial random variable.
(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 selecting the next card. This means that each card drawn has an equal probability of being a ten or non-ten, regardless of the previous outcomes.
Since the trials are independent (the outcome of one draw doesn't affect the probability of the next draw), and each trial has the same probability of success (drawing a ten), x can be considered a binomial random variable in this scenario.