99 doors to doom
You wake up in a small, locked room of a huge, luxurious castle. There are 100 doors, numbers 1 to 100, and you are to open one of them. Open the correct door, and the entire castle is yours to keep. Open any of the other 99 doors, and a hungry monster devours you instantly. There are 100 guards, one behind each door. You know that 50 of them are truth-tellers, and 50 are liars. You may ask them yes-no questions, but there's a catch: every time you ask a question, the guard will toss a biased coin, with a 0.95 probability of landing on heads. If the coin lands on tails, an alarm will be triggered and the monsters will break through the doors to devour you.
Given that you use the optimal strategy, what is the probability that you will survive and win the entire castle? Give your answer to 3 decimal places.