Final answer:
Calculating poker hand probabilities involves using combinations and the total number of possible hands. For a specific hand, the probability is 1 in 2,598,960, and for a bridge hand without a heart, it's a quotient of combinations of 39 non-heart cards taken 13 at a time over the total number of ways to choose 13 cards. Sampling with or without replacement affects the possibility of drawing the same card.
Step-by-step explanation:
Understanding Poker Hands Probabilities
The calculation of poker hands and probabilities is a mathematical problem involving combinatorics and probability theory. As an example, consider calculating the probability of getting any specific hand of five cards from a standard deck of 52 cards without replacement. There are a total of 2,598,960 different possible poker hands (calculated as 52 choose 5). The probability of getting any specific hand, therefore, is 1 in 2,598,960, which is roughly 0.0000384 or 0.00384%.
In contrast, if we are looking for the probability of being dealt a hand that does not contain a heart in a bridge game (where 13 cards are dealt), we can calculate this by considering that we need to choose 13 cards from the remaining 39 non-heart cards in the deck, resulting in combinations of 39 cards taken 13 at a time. The probability is the number of these combinations divided by the total number of ways to choose 13 cards from the full deck.
When sampling cards with replacement, the composition of the deck does not change between draws. With sampling without replacement, each draw affects the composition of the deck for the subsequent draws. Sampling with replacement allows for the possibility of drawing the same card more than once. Without replacement, as in a typical card game, each card can only be drawn once.