In a T-maze, a rate is given food if it turns left and an electric shock if it turns right. On the rst trial there is a 50-50 chance that a rat will turn either way; then, if it receives food on a given trial the probability that it will turn left on the next trial is 0.72, and if it receives a shock on a given trial the probability that it will turn left on the next trial is 0.84.

1. What is the probability that the rat will turn left on the second trial?
2. Assuming the probability of a rat turning left on any given trials depends only on what it did in the preceding trial, nd the probability that a rate will turn left on the third trial.
3. Removing the assumption in 2, calculate the expected number of trials that need to be made until the rat turns right.
4. Removing the assumption from 2 again, calculate the expected number of trials needed until the rat turns right two times in a row.