1. ## Conditional expected value

There's a mouse and three doors:

The first door takes the mouse to a cheese in 3 minutes.
The second door takes the mouse to the initial state (he has to choose a door again) in 5 minutes.
The third door takes the mouse to the initial state in 5 minutes.

Always the mouse chooses a door randomly and independient.
What's the expected value of the time the mouse takes to get the cheese?

i would really apreacciate a help here

2. ## Re: Conditional expected value

The probability of the mouse going into any of the rooms is 1/3.

denote your 3 rooms by i = 1,2,3 and the starting(initial) point by i=0.

let E(i) be the expected time till the mouse finds cheese if it has arrived at i.

now set up your equations as:

$E(0) = \frac{1}{3} E(1) + \frac{1}{3} E(2) + \frac{1}{3} E(3)$

$E(1) = 3$

$E(2)=??$

$E(3)=??$

try to figure out your equations for E(2) and E(3) and using the four equations, you can find E(0)

3. ## Re: Conditional expected value

oh! thank you! you are ideas generator machine

E(2) = 5 + E(X)
E(3) = 5+ E(X)

GREAT!

4. ## Re: Conditional expected value

Originally Posted by FRMST
oh! thank you! you are ideas generator machine

E(2) = 5 + E(X)
E(3) = 5+ E(X)

GREAT!
it should be 5+E(0)