So the probability of picking the correct configuration is 1/72 and the probability of picking the right game is 1/27 so it would be 1/1944 or 0.00051 which is better than navigating an asteroid field of 3720 to 1 .
I'm not sure how you got to those numbers. There are 18 options for each face option (eyes, hair, mouth) so the odds of picking the right face is 1/5832. And the odds of picking yahtzee are 1/26. So the actual odds of picking the right combo are 1/151632.