Suppose that two cards are randomly selected from a standard 52-card deck. (a) What is the probability that the first card is a queen and the second card is a queen if the sampling is done without replacement? (b) What is the probability that the first card is a queen and the second card is a queen if the sampling is done with replacement? (a) If the sampling is done without replacement, the probability that the first card is a queen and the second card is a queen is $\square$ (Round to three decimal places as needed.)
Answer (1)
Without replacement, the probability of drawing two queens is calculated as 52 4 × 51 3 .
With replacement, the probability of drawing two queens is calculated as 52 4 × 52 4 .
The probability without replacement is approximately 0.005.
The probability with replacement is approximately 0.006. 0.005
Explanation
Problem Analysis We are asked to find the probability of drawing two queens in a row from a standard 52-card deck, first without replacement and then with replacement. Let's break down each scenario.
Calculate Probability Without Replacement (a) Without Replacement: When drawing without replacement, the first card drawn is not returned to the deck. This affects the probabilities for the second draw.
Probability of the first card being a queen: There are 4 queens in a 52-card deck, so the probability is 52 4 .
Probability of the second card being a queen, given the first card was a queen: After drawing one queen, there are only 3 queens left, and only 51 cards in the deck. So the probability is 51 3 .
To find the probability of both events happening, we multiply the probabilities: 52 4 × 51 3 = 2652 12 = 221 1 ≈ 0.00452 Rounding to three decimal places, we get 0.005.
Calculate Probability With Replacement (b) With Replacement: When drawing with replacement, the first card drawn is returned to the deck before the second card is drawn. This means the probabilities for the second draw are the same as the first draw.
Probability of the first card being a queen: There are 4 queens in a 52-card deck, so the probability is 52 4 .
Probability of the second card being a queen: Since we replace the first card, there are still 4 queens in a 52-card deck, so the probability is 52 4 .
To find the probability of both events happening, we multiply the probabilities: 52 4 × 52 4 = 2704 16 = 169 1 ≈ 0.00592 Rounding to three decimal places, we get 0.006.
Final Answer Therefore, the probability of drawing two queens without replacement is approximately 0.005, and the probability of drawing two queens with replacement is approximately 0.006.
Examples
Understanding probabilities with and without replacement is crucial in many real-world scenarios, such as quality control in manufacturing. For example, if a factory produces items and tests a sample, knowing whether the tested items are returned to the production line (replacement) or not (without replacement) affects the probability calculations for finding defective items in the next test.
