A psychologist conducts an experiment in which a mouse is placed in a T-maze, where it has a choice at the T-junction of turning left and receiving a reward (cheese) or turning right and receiving a mild shock. At the end of each trial a record is kept of the mouse's response. It is observed that the mouse is as likely to turn left (state 1) as right (state 2) during the first trial. In subsequent trials, however, the observation is made that if the mouse had turned left in the previous trial, then the probability that it will turn left in the next trial is .6, whereas the probability that it will turn right is .4. If the mouse had turned right in the previous trial, then the probability that it will turn right in the next trial is .2, whereas the probability that it will turn left is .8. In the long run, what percentage of the time will the mouse turn left at the T-junction? Please round the answer to one decimal place.
A) 23.2%
B) 66.7%
C) 93.0%
D) 50.4%

Question

A psychologist conducts an experiment in which a mouse is placed in a T-maze, where it has a choice at the T-junction of turning left and receiving a reward (cheese) or turning right and receiving a mild shock. At the end of each trial a record is kept of the mouse's response. It is observed that the mouse is as likely to turn left (state 1) as right (state 2) during the first trial. In subsequent trials, however, the observation is made that if the mouse had turned left in the previous trial, then the probability that it will turn left in the next trial is .6, whereas the probability that it will turn right is .4. If the mouse had turned right in the previous trial, then the probability that it will turn right in the next trial is .2, whereas the probability that it will turn left is .8. In the long run, what percentage of the time will the mouse turn left at the T-junction? Please round the answer to one decimal place. ​
A) 23.2%
B) 66.7%
C) 93.0%
D) 50.4%

Quizplus · Accepted Answer

The mouse's behavior is described by a Markov chain. We can find the long-run probabilities by setting up and solving a system of equations. Let P(i,j) denote the probability of transitioning from state i to state j. Then the following system of equations holds:

P(1,1) = 0.6P(1,1) + 0.2P(2,1)
P(1,2) = 0.4P(1,1) + 0.8P(2,1)
P(2,1) = 0.4P(1,2) + 0.8P(2,2)
P(2,2) = 0.6P(1,2) + 0.2P(2,2)
P(1,1) + P(1,2) = 1
P(2,1) + P(2,2) = 1

Solving this system of equations, we get P(1,1) = 0.667 (rounded to three decimal places). Therefore, in the long run, the mouse will turn left at the T-junction 66.7% of the time. The answer is B.

A Psychologist Conducts an Experiment in Which a Mouse Is