Problem 7

Ralph's alarm clock goes off at 6.15am, and he reaches out into the darkness to turn it off.

He is successful with probability $1/2$. But he sometimes misses and instead knocks over either his bedside light or his glass of water, each with probability $1/4$ — and in this case he now needs to right the knocked over item as well as turn off the alarm.

To be precise, each of three items on the bedside table has two states (alarm on or off, light upright or knocked over, glass upright or knocked over). Ralph wants to set the state to (alarm off, light upright, glass upright) so that he can go back to sleep, and until this is the case, he will repeatedly:

choose with equal probability one of the items that is not in the desired state
attempt to correct the state of that item, which he does successfully with probability $1/2$. However, with probability $1/4$ he instead flips the state of each one of the other two items.

What's the probability that Ralph manages to go back to sleep without spilling his water?

Give your answer to three decimal places