Final answer:
The question delves into the realm of probability and combinatorics using a standard deck of 52 cards, exploring microstates, specific card-hand probabilities, and expected profits in card guessing games. Calculations for microstates and probabilities are based on standard combinatorial formulas and understanding of independent events.
Step-by-step explanation:
Card Combinations and Probability Calculations
When dealing with problems in probability and combinatorics, a standard deck of 52 cards is often used for illustration. Let's explore a few scenarios that exemplify these concepts:
- Microstates in Card Draws: Drawing five random cards from five separate decks—each deck leading to 52 possible outcomes—results in 52^5 microstates. This is because each card drawn is an independent event with 52 outcomes.
- Probability of 5 Queens of Hearts: Getting the queen of hearts from five separate decks, one from each, has the probability of (1/52)^5 as each draw is independent and has the probability of 1/52.
- Probability of a Specific Hand: The likelihood of getting any specific five-card hand from a single deck is calculated by the combination formula, (52 choose 5), which equals 1 in 2,598,960.
Permutations and combinations play a crucial role in calculating probabilities. For example, calculating combinations using factorial notation (e.g., 4!) helps in understanding how many different ways something can occur. The permutation of four different suits is calculated as 4! = 4 x 3 x 2 x 1 = 24 different combinations.
Regarding expected profit in card games, it's essential to consider the payout structure and the likelihood of each outcome. For instance, if you guess the suit of four cards drawn with replacement and win only if all guesses are correct, the profit expectation relies on the probability of guessing correctly (1/4)^4 and the payout, which in this scenario is $256 for a $1 game.