In a game show the contestants are required to search and hit six buzzers which are placed inside a hexagonal hall having six gates. The below given figure gives an idea about the hall but the contestants are unaware about the structure of the hall. They just know that they can enter and exit from any of the six gates. The first three contestants are good friends and decide which doors they will use and latch those doors in order to trouble other contestants. The first contestant enters from a door, latches it from inside, hits the buzzers and leaves from another door latching it from outside. The same is done by the other 2 contestants. The 4th
contestant is a nice person, he enters by opening a door latched from outside, leaves it unlatched, hits the buzzers and leaves by opening a door latched from inside without latching it from outside. The 5th
contestant comes to know that some doors are latched, hence he even latches the doors which he uses i.e. if he comes across a latched door and he can open it he uses it and latches it from other side and if he moves through an unlatched door he latches it after using it. Now the 6th
contestant starts. If he picks a door at random to enter what is the probability that he will be able to enter through that door?