Final answer:
The probability of selecting the king of hearts from a standard deck of 52 cards is 1/52. For the other scenarios, the probability calculations differ based on whether the selection is done with or without replacement, using combinations.
Step-by-step explanation:
The probability of selecting the king of hearts from a well-shuffled standard deck of 52 cards is straightforward to calculate. There is only one king of hearts in a deck of cards and 52 possible cards to choose from. Hence, the probability is 1 out of 52, or 1/52.
When addressing the calculation on the probability of being dealt a hand without a heart, we must consider combinations since the order in which the cards are dealt is not important.
We want to find the number of ways to select 13 cards without picking any hearts, which means we choose from the 39 cards that are not hearts (13 spades, 13 diamonds, and 13 clubs).
This can be calculated using the combination formula C(n, k) = n! / (k!(n-k)!), where n is the total number of items to pick from, and k is the number of items to choose. The probability then is C(39, 13) divided by C(52, 13).
For Try It Σ 3.5, possible outcomes are dependent on whether the sampling is with or without replacement. With replacement, each card draw is independent, and cards can be drawn repeatedly. Without replacement, each card is unique and cannot appear more than once in the sampled hand of cards.
Only outcomes a and c are possible when you sample without replacement, since each card is unique. Outcome b can only happen with replacement, as the king of hearts appears twice, which is not possible without replacement.