I need help with a statistic problem. The problem is:
A room has 3 doors. One door leads to the room's exit in 0 minutes, the second door will lead us back to the same room after 3 minutes, and last door will return us to the room after 6 minutes.
After opening one of doors we don't remember which one was it. In other words, the probability to choose each door is always $\dfrac13$. (We choose a door randomly each time.)
What is the average time in minutes to exit from the room?
It says the answer is 10, but don't know how they got it.
Thanks a lot!
You are looking for the expected value of the time to exit. If $t$ is the expected time, you have $1/3$ chance of choosing each door. If you choose one of the ones that does not exit, you are back where you started, so $t=\frac 13 \cdot 0 + \frac 13 (t+3) + \frac 13(t+6)$ From this I get $9$, not $10$, minutes.