Suppose that Willa encounters two traffic lights on her commute to work. Based on her past experience, she judged that the probability is 0.35 that the first light will be red when she gets to it, 0.45 that the second light will be red, and 0.60 that at least one of the lights will be red. Let R1 denote the event that the first light is red, and let R2 denote the event that the second light is red.
a. Determine the probability that both lights will be red. Justify your answer, using a probability formula or a Venn diagram.
b. Find the probability that neither light will be red.
c. Determine the conditional probability that the second light will be red given that the first light is red. (Show how you calculate this.)
d. Are the events R1 and R2 independent? Justify your answer numerically.
e. Are the events R1 and R2 disjoint (mutually exclusive)? Justify your answer numerically.