Use Bayes Theorem to solve the following:

1. It's a typically hot morning in July in Delhi. You look outside and see some dark clouds rolling in. Historically, there is a 30% chance of rain on any given day in July. Furthermore, on days when it does in fact rain, 95% of the time there are dark clouds that roll in during the morning. But, on days when it does not rain, 25% of the time there are dark clouds that roll in during the morning. Is it going to rain today?

2. When IRS receives tax forms, it puts them through a computer to flag forms that need to be investigated further. The computer looks for mistakes in the forms, for example addition mistakes or incorrect deduction amounts. Suppose the computer correctly flags 80% of all returns that have mistakes, and it incorrectly flags 5% of error-free returns. Further, suppose that 15% of all tax returns have errors. A tax return is flagged by the computer. What is the chance that it contains mistakes, given that the computer flagged it?