Howard made a simple robot. This robot is placed in the space outside a maze consisting of three rooms and six doors, as depicted in the following figure. Figure 1: The layout of the maze. 1 2 3 Whenever the robot is in a space or room with 𝑘 doors, it chooses each of these doors to move through next with probability 1/𝑘 . Suppose the robot is in Room 1, find the probability of it leaving the maze (for the first time) from Room 3.
Question
Howard made a simple robot. This robot is placed in the space outside a maze consisting of three rooms and six doors, as depicted in the following figure. Figure 1: The layout of the maze. 1 2 3 Whenever the robot is in a space or room with 𝑘 doors, it chooses each of these doors to move through next with probability 1/𝑘 . Suppose the robot is in Room 1, find the probability of it leaving the maze (for the first time) from Room 3.
Solution
To solve this problem, we need to understand the movement of the robot. The robot is currently in Room 1 and it has two doors to choose from. It can either go to Room 2 or directly to the outside.
Step 1: The probability of the robot going to Room 2 from Room 1 is 1/2 (since there are two doors in Room 1).
Step 2: Once in Room 2, the robot has three doors to choose from. It can either go back to Room 1, go to Room 3, or go directly to the outside. The probability of going to Room 3 from Room 2 is 1/3 (since there are three doors in Room 2).
Step 3: Once in Room 3, the robot has two doors to choose from. It can either go back to Room 2 or go directly to the outside. The probability of going outside from Room 3 is 1/2 (since there are two doors in Room 3).
So, the total probability of the robot leaving the maze from Room 3 is the product of the probabilities at each step.
Probability = (1/2) * (1/3) * (1/2) = 1/12.
So, the probability of the robot leaving the maze (for the first time) from Room 3 is 1/12.
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