You are a medical claims processor for Humana. Claims arrive at your desk at an average rate of three claims per day, but the actual number of claims fluctuates from day to day. You believe that the elapsed times between consecutive claims are exponentially distributed. You also know that each claim has a 2/3 chance of being from a female patient, independently of the status of all other claims. The amount paid on a claim from a female patient is random with a mean of $1000 and a standard deviation of $500.
What is the probability that you will pay at least $55,000 in total claims for female patients during the next 30 days?