A prisoner is placed in a circular room with six doors around the room numbered clockwise 1 through 6. He is told that behind two consecutive doors (ie 1,2 or 2,3 or 3,4 or 4,5 or 5,6 or 6,1) are lions. If the prisoner is able to open two doors without releasing a lion, he will be set free. Otherwise he will be lunch. However, there is a catch. His first choice depends upon the roll of a die. If he survives his first choice, for his second choice he may either select the next door in line (eg if he rolled a 3 first he may subsequently choose door 4), or he may roll the die again.
Now suppose he is successful with his first door. That is he rolls the die, opens the door according to the number rolled and there is no lion behind the door. For his next choice, should he select the next door in line, or should he roll the die again, and why? Note that it is permissable for the same door to be opened twice. For example, if the prisoner rolls a 3 to begin and opens door 3 successfully he may either choose door 4 as his second choice or roll the die again in order to select another door. He is allowed to roll another 3 which of course would mean he was successful again. You should also assume that the die is fair. Each number is equally likely.
I don't understand how to get the probability of getting a safe door by choosing the next door in line. I asked my teacher and he tried to help, but then had to be summoned elsewhere. I did get a hint, I don't understand what he said. The thing he told me was that we know he is successful with the first door, so be entered one of the safe doors. So, now only 1 of the 4 safe doors has a bad door next to it, which is a 1/4 chance of getting a bad door (or 3/4 chance of getting another safe door). This is what I'm confused about, how we get down to 4 doors? I thought it was a 3/5 chance since 3 of the 5 doors are safe (still only 2 doors have lions behind them).
I did figure out that if he decided to roll the die again, he will have a 4/6 chance (2/3) of getting a safe door, and a 2/6 chance (1/3) of getting a bad door, since 2 of the 6 doors have lions behind them.