Final answer:
The probability of drawing 8 face cards from a deck without replacement requires complex calculations using combinatorics. Determining sampling replacement can be inferred from duplicates drawn: drawing two cards of the same value, e.g., Q of spades, implies sampling with replacement.
Step-by-step explanation:
Finding the Probability of Drawing 8 Face Cards without Replacement:
To calculate the probability of drawing 8 face cards from a deck without replacement, we first need to know the number of face cards in a standard deck of 52 cards. There are 12 face cards in a deck, comprising of kings, queens, and jacks from four suits.
The probability of drawing the first face card is 12/52, since there are 12 face cards in the deck. After drawing one card, the probability of drawing another face card decreases because the number of face cards and the total number of cards in the deck both decrease by 1. The calculation for the probability of drawing each additional face card must account for the lower numbers.
However, calculating the probability of drawing 8 face cards in succession without replacement is extremely complex and beyond the scope of a simple explanation. It would require the use of combinatorics and the hypergeometric distribution.
Regarding whether we can decide if the sampling was with or without replacement given specific cards drawn, consider this:
- If we have a Q of spades, K of hearts, and another Q of spades, the sampling could not be without replacement because a standard deck of cards has only one Q of spades. This must be with replacement.
- If we have a Q of spades, K of hearts, and J of spades, the sampling can be without replacement, because these cards are different and can be drawn from the deck only once.
For the case with the sampling of four cards from a deck:
- QS, 1D, 1C, QD - This outcome is possible without replacement.
- KH, 7D, 6D, KH - This requires replacement since the KH is drawn twice.
- QS, 7D, 6D, KS - This outcome is possible without replacement.