1. Two friends (A and B) agree to meet on 4:00 PM. A usually arrives between 5 minutes early and 5 minutes late. B usually arrives between 5 minutes early and 15 minutes late. Their times of arrival are independent from each other.

    1. What is the probability that B arrives definitely later than A?
    2. What is the expected time that A waits B?
    3. What is the probability that both meet early?

  1. There are three computers, which provides answers to questions with speed according to exponential distribution with means (\(1/\lambda\)) 6, 4 and 3 per hour, respectively. What is the probability that at least one machine provides an answer within the first hour?

  2. Time between customer arrivals in a cafe is exponential with the mean value of 6 minutes.

    1. What is the probability that no customers arrive in 15 minutes?
    2. What is the interarrival time if the probability of a customer to arrive is 0.9?
    3. What is the probability that 10 customers arrive in the first hour?
    4. What is the probability of getting the first customer in 15 minutes if no customer arrived in the first 10 minutes?

    Hint: Check the relationship between Poisson and Exponential distributions.

  1. A pack of flour contains 1 kg of flour. Though a flour pouring machine has a standard deviation of 50 gr.

    1. What is the probability that a randomly selected package contains between 925-1075 grams of flour?
    2. If a proper flour package should contain between 1000-x and 1000+x grams of flour, what should x be that 80% of the packages are deemed proper?
    3. Your customer strictly declared that 95% of the packages should contain at least 1000 grams of flour, so you should adjust the mean value. What should be the new mean value?

  2. There are two different roads to get to Sarıyer. Road A takes 35 minutes on average with standard deviation 5 minutes. Road B takes 32 minutes on average with standard deviation 8 minutes.

    1. Which road has the higher advantage if one wants to reach Sarıyer in 42 minutes?
    2. What is the maximum time of arrival with 90% probability? Calculate for each road.