Question

A rat is trapped in a maze. Initially it has to choose one of two directions. If it goes to the right, then it will wander around in the maze for three minutes and will then return to its initial position. If it goes to the left, then with probability \(\frac{1}{3}\) it will depart the maze after two minutes of traveling, and with probability \(\frac{2}{3}\) it will return to its initial position after five minutes of traveling. Assuming that the rat is at all times equally likely to go to the left or the right, what is the expected number of minutes that it will be trapped in the maze?

Short answer

The expected number of minutes the rat will be trapped in the maze is 4.5 minutes.

Step-by-step solution

Identify the possible outcomes and their probabilities

The rat has the following possible outcomes and probabilities: - Go right with a 1/2 probability, return to initial position after 3 minutes - Go left, and then exit after 2 minutes with a 1/2 * 1/3 = 1/6 probability - Go left and return to initial position after 5 minutes with a 1/2 * 2/3 = 1/3 probability

Calculate the expected time for each possible outcome

Now we need to calculate the expected time for each possible outcome: - For the first outcome (going right): The rat will spend 3 minutes wandering and then return to its initial position. So the expected time for this outcome is 3 minutes. - For the second outcome (going left and exiting): The rat will spend 2 minutes traveling and then exit the maze. So the expected time for this outcome is 2 minutes. - For the third outcome (going left and returning): The rat will spend 5 minutes wandering and then return to its initial position. So the expected time for this outcome is 5 minutes.

Calculate the overall expected time for the rat to be trapped in the maze

To find the overall expected time, we will multiply the expected time of each outcome by their respective probability, and then sum up these products: Expected time = (1/2 * 3) + (1/6 * 2) + (1/3 * 5) Expected time = (3/2) + (1/3) + (5/3) Expected time = \(\frac{3+1+5}{2}\) = \(\frac{9}{2}\) = 4.5 minutes Hence, the expected number of minutes the rat will be trapped in the maze is 4.5 minutes.