Final answer:
The questions involve determining the number of possible outcomes, calculating combinations for specific outcomes, and finding probabilities for these outcomes in the context of flipping 50 fair coins.
Step-by-step explanation:
When flipping coins, we are dealing with the concepts of probability and combinatorics. In part (a), to determine the number of microstates for 50 fair coin flips, we use the formula for the total number of outcomes of independent events: 2n, where n is the number of coins flipped. Since each coin has 2 possible outcomes (heads or tails), for 50 coins, there would be 250 possible microstates.
For part (b), finding the number of ways to achieve exactly 25 heads and 25 tails is equivalent to computing a combination, specifically 50 choose 25, denoted as C(50, 25), which can be calculated using the combination formula. The combination formula for choosing k items from n items is given by n! / [k!(n-k)!].
For part (c), the probability of getting exactly 25 heads and 25 tails is the number of ways to achieve that particular outcome (calculated in part b) divided by the total number of possible outcomes (calculated in part a).
Lastly, in part (d), the probability of this specific outcome of 30 heads and 20 tails is similar to that in part (c), but we would use C(50, 30) for our numerator as we are now looking at the combination of 50 coins taken 30 at a time.