Probability problem

[enthumath]

Hey I'm having trouble with this math question,

Rachael purchases a townhouse 'off the plan' and the builder offers her a choice of colors for interior painting. The walls can be cream, light blue or light green, and the ceilings vivid white or off-white. Different colors may be chosen for different rooms but only two colors can be used in any room (one for the ceiling and one for the walls). The townhouse consists of four rooms; living room, bedroom, kitchen and bathroom.

What is the probability that at least 2 rooms have identical color schemes.

Thanks for the help

[Rassis]

A Monte Carlo simulation model built in Excel yields approximately 0,73.

[Logophobic]

Exact probability: 13/18

There are 6 different color schemes, with presumably equal chance of occuring in any given room. Taking the rooms in any order, we can easily figure that there is a 1 in 6 chance that the first two rooms match. If they don't match, look at the third room. There is a 1 in 3 chance that this room will match one of the first two. If there is still no match, look at the last room. There is a 1 in 2 chance that it matches one of the other three.

Taking into account the probability that the later checks need to be made, we have:
1/6 + 5/6 * 1/3 + 5/6 * 2/3 * 1/2 = 3/18 + 5/18 + 5/18 = 13/18

It is perhaps easier to determine the probability that there are no matching rooms:

6/6 * 5/6 * 4/6 * 3/6 = 5/6 * 2/3 * 1/2 = 5/6 * 1/3 = 5/18

This then leaves us with 13/18 for at least one match.