Harry Potter in The Magical Maze

In the scenario there is no memory, assume that E is the escape time. Given equal probability of each of the 4 exits, one can formulate the following equation:

E = (E + 8)/4 + (E + 2)/4 + (3 + 5)/4

Solving for E gives 9 minutes = 540 seconds.

Random simulations:

Python 2.7:

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import random
def walk():
    time = 0
    while True:
        path = random.choice('abcd')
        if path == 'a':
            time += 5
            return time
        elif path == 'b':
            time += 8
        elif path == 'c':
            time += 3
            return time
        elif path == 'd':
            time += 2
total_time = 0.0
trials = 1000000
for trial in xrange(trials):
    total_time += walk()
print "Minutes:", total_time/trials

The probability of choosing each one of the tracks is $\frac 14$ . Let $T$ be the expected time to reach the end of the maze. If he chooses Path A , he would reach in $5$ minutes, if he chooses C, $3$ minutes, if he chooses B, then $8 + T$ minutes, since he goes back to the start again, and D $2+ T$ minutes.

Then total expected time $T = \frac 14 (5 + (8+T) + 3 + (2+T) )\Rightarrow T = 9 \text{ minutes}$