The number of bugs in a computer program is a random variable, which follows a Poisson distribution with expected value 12.5. After extensive testing and debugging for a week, each bug will independently be discovered and removed with probability 0.8. Suppose that after the week-long testing and debugging, we have discovered and removed 6 bugs. What is the probability that there are still 3 or more bugs remaining in the computer program?

Trains arrive at a major train-station randomly and independently; the probability of an arrival is the same for any two time intervals of equal length. The mean arrival rate is 4 trains per hour. What is the probability that 8 trains will arrive during any given hour of operation?

suppose we select an SRS of 1010 adults. If it is believed that 60% of all adults are very satisfied with their lives, what is the probability that at least 610 adults in our sample are very satisfied with their lives?Let X be the number (out of 1010) of adults who are very satisfied with their lives. What distribution should X have? Why?

After careful research you observe that the number of hackers active on each day is a random variable Bin(3,0.5) distribution. If no hackers are active, the probability of the website failure is 0.15; If one hacker is active the probability of the website failure is 0.3; if two or more hackers are active the probability of the website failure is 0.5. You conclude that the total probability of website failure on a given day is two decimal place). Assuming there is a failure, the probability no hackers are active is (to (to two decimal places).