Aaron is new to a city. According to his experience in his hometown, Aaron initially believes that September is spring. So he believes that for a random day in September, the probability it is hot is 20%, the probability it is warm is 70%, and the probability it is cold is 10%. Assume the weather of each day is independent.

1. What does Aaron initially believe is the probability of having 5 straight cold days in September?

2. After arriving in the new city and experiencing some really cold weather, Aaron starts to wonder if maybe the seasons work differently there. He still thinks September is spring with 90% confidence, but acknowledges a 10% chance that somehow September is still the dead of winter in the new city. Aaron believes that in the winter, the probability of a hot day is 0%, the probability of a warm day is 30%, and the probability of a cold day is 70%.

Over the next 5 days, Aaron observes cold, warm, cold, cold, warm. After these observations, what does Aaron now believe is the probability that it is spring in September?