Final answer:
The question involves logic puzzles and probability calculations in mathematics, requiring an understanding of conditions for placing letters in a grid and the calculation of probabilities based on given events and a defined sample space.
Step-by-step explanation:
The question seems to be a mix of logic puzzle and probability concepts, suggesting it pertains to mathematics, likely at the high school level. To determine the correct letters for Mattew and Holliday's shaded squares, we need to adhere to the rules given. For the grid puzzles, we have a set of conditions to follow, such as any shaded square must contain the same letter as the square directly above it, and any unshaded square must contain a different letter from the square directly above it. For example, if a row ends with the letters 'ld', the word above must end with 'ea', leading to possible word combinations like 'beat' and 'bald' or 'meat' and 'mild'.
When addressing the probability component, we work with different events and calculate the probability of these events occurring given a sample space. For instance, let H be the event of drawing a hat from a party favor box. The probability of this event would involve the number of hats over the total number of items in the box.
In terms of probability with sample spaces and events, we can define the sample space S, determine the probability of events like A (even numbers) and B (numbers greater than 13), and find the probability of the intersection between these events (A AND B).