Final answer:
The probability that the first card drawn is an ace and the second is a queen without replacement from a standard 52-card deck is 1/166. If identical cards are drawn, the sampling was with replacement. If there are no repeats in the drawn cards, then the sampling was without replacement.
Step-by-step explanation:
To answer the question about the probability that the first drawn card is an ace and the second is a queen when drawing without replacement from a standard 52-card deck, we must calculate the probability of each event occurring consecutively, without the first card returning to the deck. Initially, there are 4 aces and 52 cards in total. The probability of drawing an ace first is therefore 4/52. Once an ace is drawn, there are now 51 cards left in the deck, which includes 4 queens. The probability of then drawing a queen is 4/51.
The combined probability of both events happening in succession is found by multiplying the two probabilities together:
- P(First card is an ace) = 4/52
- P(Second card is a queen given first is an ace) = 4/51
The combined probability of drawing first an ace and then a queen without replacement is:
P(Ace first, Queen second) = P(First card is an ace) × P(Second card is a queen given first is an ace) = (4/52) × (4/51)
Computing this, we get the probability as:
P(Ace first, Queen second) = 16/2652
After simplification, this can be reduced to:
P(Ace first, Queen second) = 1/166
As for the information provided about drawing cards and determining whether the draws were with or without replacement, the following observations can be made:
a. If you draw the Q of spades, K of hearts, and then the Q of spades again, this implies that sampling must have been done with replacement, because the same card was drawn twice.
b. If you pick the K of hearts, then the three of diamonds, and finally the J of spades without seeing the same card twice, this suggests that the sampling was done without replacement.