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?
pr=4/52 * 3/51
b)= 4/52 * 4/52
To solve this problem, we need to calculate the probabilities separately for sampling without replacement and sampling with replacement.
(a) Probability of drawing a queen from a standard 52-card deck is 4/52 since there are 4 queens in a deck, and the number of cards is 52.
To calculate the probability of drawing another queen without replacement, we have 3 queens left in the remaining 51 cards.
Therefore, the probability of drawing a queen as the second card without replacement is 3/51.
Since both events are independent, we can multiply the probabilities together:
P(1st card is queen and 2nd card is queen without replacement) = (4/52) * (3/51)
(b) When sampling is done with replacement, after drawing the first queen, we place the card back into the deck. So, in each draw, we have the same probability of drawing a queen.
The probability of drawing a queen with replacement is 4/52 for both the first and second cards.
So, P(1st card is queen and 2nd card is queen with replacement) = (4/52) * (4/52)
Note: You can simplify the fractions if you wish.
I hope this helps! Let me know if you have any further questions.
To solve this problem, we need to consider two different scenarios:
(a) Sampling without replacement:
In this case, after the first card is chosen, there will be 51 cards remaining in the deck, with 3 of them being queens. To determine the probability of the second card being a queen, we need to divide the number of favorable outcomes (selecting a queen) by the number of possible outcomes (choosing any card from the remaining 51 cards):
Number of favorable outcomes: There is 1 queen left in the deck.
Number of possible outcomes: There are 51 cards remaining.
Therefore, the probability of the second card being a queen, given that the first card is a queen (without replacement), is: 1/51.
(b) Sampling with replacement:
In this case, after the first card is chosen, it is put back into the deck. This means that there will still be 4 queens remaining in the deck of 52 cards for the second draw. The probability of drawing a queen remains the same for both draws since we are replacing the card each time.
Number of favorable outcomes: There are 4 queens left in the deck.
Number of possible outcomes: There are still 52 cards in the deck.
Therefore, the probability of the second card being a queen, given that the first card is a queen (with replacement), is: 4/52, which simplifies to 1/13.
To summarize:
(a) Probability of the second card being a queen, given that the first card is a queen (without replacement): 1/51.
(b) Probability of the second card being a queen, given that the first card is a queen (with replacement): 1/13.