perplexus.info :: Probability : The mouse and the cheese

The mouse and the cheese (Posted on 2008-09-11)

A mouse has 3 rooms to go into.

If it goes into room 1, it will find the cheese after 3 minutes.

If it goes into room 2, it will look for cheese for 4 minutes, won't find it, and will go out.

If it goes into room 3, it will look for cheese for 5 minutes, won't find it, and will go out.

The mouse will not remember that it was in rooms 2 and 3 after it goes out of them, and it will continue going in and out until it finds the cheese. (It can go into the same room again and again.)

What is the average time for the mouse to find the cheese?

Submitted by pcbouhid

Rating: 3.0000 (2 votes)

Solution: (Hide)

This is sure an old problem, but the mouse is younger than that and the cheese is fresh.

To simplify, if we combine the rooms b and c, we have a time of 4.5 and probability 2/3 for them, i.e.

(1/3)(4) + (1/3)(5) = (1/3)(9) = (2/3)(4.5)

Now, we can construct a table, calling f the combined rooms b and c:
Sequence Time (mins) Probability -------- ----------- ----------- a 3 1/3 fa 7.5 (2/3)(1/3) f^2.a 12 (2/3)^2 (1/3) f^3.a 16.5 (2/3)^3 (1/3) f^4.a 21 (2/3)^4 (1/3)
and so on to infinity.

The expected time is:

T = 3*(1/3) + 7.5*(2/3)(1/3) + 12*(2/3)^2 (1/3) + ...
= (1/3)*[3 + 7.5*(2/3) + 12*(2/3)^2 + 16.5*(2/3)^3 + ...]

Now to sum the series in brackets we represent (2/3) by x and we have:

S = 3 + 7.5x + 12x^2 + 16.5x^3 + 21x^4 + ...
xS = 3x + 7.5x^2 + 12x^3 + 16.5x^4 + ...

(1-x)S = 3 + 4.5x + 4.5x^2 + 4.5x^3 + 4.5x^4 + ...

(1-x)S = 3 + 4.5x(1 + x + x^2 + x^3 + ...)

(1-x)S = 3 + 4.5x(1/(1-x))

S = 3/(1-x) + 4.5x/(1-x)^2

and putting x = 2/3, 1-x = 1/3:

T = (1/3)*[3/(1/3) + 4.5(2/3)/(1/3)^2]

= (1/3)*[9 + 27]

= 12 minutes

q.e.d. - Nadav Shalit

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Old Problem	ed bottemiller	2008-09-11 17:03:16
re: solution	xdog	2008-09-11 15:54:14
re: solution	Charlie	2008-09-11 13:56:10
solution	xdog	2008-09-11 12:17:51

Please log in:

Login:
Password:
Remember me:

Sign up! \| Forgot password

Search:

Search body:

Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (6)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:


perplexus dot info