6. [6] Sarah is deciding whether to visit Russia or Washington, DC for the holidays. She makes her decision by rolling a regular 6 -sided die. If she gets a 1 or 2 , she goes to DC. If she rolls a 3,4 , or 5 , she goes to Russia. If she rolls a 6 , she rolls again. What is the probability that she goes to DC?
Answer: $\quad \boxed{\frac{2}{5}}$ On each roll, the probability that Sarah decides to go to Russia is $3 / 2$ times the probability she decides to go to DC. So, the total probability that she goes to Russia is $3 / 2$ times the total probability that she goes to DC. Since these probabilities sum to 1 (they are the only two eventual outcomes) Sarah goes to DC with probability $\frac{2}{5}$ and Russia with probability $\frac{3}{5}$.