carcass wrote:
Mike and Emily need to build 2 identical houses. Mike, working alone, can build a house in 6 weeks. Emily, working alone, can build a house in 8 weeks. To determine who will do the building they will roll a fair six-sided die. If they roll a 1 or 2, Mike will work alone. If they roll a 3 or 4, Emily will work alone. If they roll a 5 or 6, they will work together and independently. What is the probability both houses will be completed after 7 weeks?
A) 0
B) 1/3
C) 1/2
D) 2/3
E) 1
Cool question - a lot going on here.
I'm going to take a shortcut based on some logic:
The only way for the 2 houses to get done in under 7 weeks is if they work together. If Mike works alone - it would take him 12 weeks to build 2 houses. If Emily works alone, it would take 16 weeks. The check for this is below.
Together, they have a rate of \(\frac{6*8}{6+8}\) per house. Knowing that they'll need 2, we get \(2*(\frac{6*8}{6+8})=6 \frac{6}{7}\)
Knowing this - the only way to complete in under 7 weeks is to work together - we can move on to the probability. This is very simple: 2 sides of the dice (a 5 or a 6) out of 6 possible outcomes is 2/6, which reduces to
1/3.. The Answer is B.