The Chicago Art Museum's Renaissance display consists of four hallways bordered around a square courtyard. A single guard is assigned to patrol the four hallways. When the guard starts working, he begins in one of the corners and walks clockwise. When he arrives at a subsequent corner he flips two coins. If both coins are heads, he changes the direction he is walking. Otherwise, he continues in the same direction. Let E be the expected number of lengths of hallway that he walks before he first returns to his starting corner. Let p be the probability that he walks strictly more than E lengths of hallway before returning to his starting corner. p can be expressed as ab where a and b are coprime positive integers. What is the value of a+b?

The guard may walk in the same hallway more than one time. Each time he walks in it counts as one length of hallway.

25 is it

its 5

nooooo....its wrong

To solve this problem, we need to find the expected number of lengths of hallway that the guard walks before he first returns to his starting corner.

Let's consider each hallway separately. Since all four hallways are identical, it doesn't matter which one we analyze. We can assume he starts at that specific hallway.

Let's say the length of the hallway is "L". The guard will either walk "L" lengths of hallway and return to the starting corner or walk more than "L" lengths of hallway before returning.

To determine the expected number of lengths of hallway he walks before returning, we need to consider the probability of each scenario.

Case 1: The guard walks exactly "L" lengths before returning.
In order for the guard to return to the starting corner, he must walk through every hallway, including the starting hallway, at least once. This means he must walk through the other three hallways before returning. The probability of this happening is (1/2)^3 = 1/8 because he flips two coins at each subsequent corner and gets heads on both coins (1/2 * 1/2 * 1/2 = 1/8).

Case 2: The guard walks more than "L" lengths before returning.
In this scenario, the guard has walked through the other three hallways and returned back to the starting hallway without returning to the starting corner. The probability of this happening is 1 - 1/8 = 7/8.

Now, let's calculate the expected number of lengths of hallway the guard walks:

E = P(case 1) * "L" + P(case 2) * (E + "L")

Since the guard starts in the hallway and moves to the subsequent corners in a clockwise direction, we can ignore the direction change condition for now.

Simplifying the equation, we get:

E = (L/8) + (7/8)(E + L)

Solving this equation for E, we get:

E = 7L/8 + 7E/8 + 7L/8

8E - 7E = 7L/8 + 7L/8

E = 14L/8 = 7L/4

Now, we need to consider the direction change condition. Let's denote the probability of direction change as "p".

The guard walks through each hallway and flips two coins at each subsequent corner. There are 4 corners, so there are 2^4 = 16 possible outcomes for the coin flips. Out of these 16 outcomes, there are 2 outcomes where both coins are heads, resulting in a direction change.

Therefore, p = 2/16 = 1/8.

Finally, we need to calculate the probability that he walks strictly more than E lengths of hallway before returning to his starting corner.

Since the guard walks exactly E lengths of hallway before returning with a probability of (1/8), the probability of walking strictly more than E lengths is (7/8).

Therefore, p = 7/8.

The value of a+b = 7+8 = 15.

So, the answer to the question is 15.