1. Accuracy in taking orders at a drive-through window is important for fast-food chains. Periodically, QSR Magazine publishes "The Drive-Thru Performance Study: Order Accuracy" that measures the percentage of orders that are filled correctly. In a recent month, the percentage of orders filled correctly at Burger King was approximately 80%. Suppose that you and three friends go to the drive-through window at Burger King and place an order. Three friends of yours independently place orders at the window at the same store. (please write calculation process) a. [2pt] What is the probability that two of the four will be filled correctly? b. [2pt] What is the probability that at least one of the four will be filled correctly? c. [2pt] What is the probability that only your order will be filled correctly? d. [1pt] What is the mean of the binomial distribution for the number of orders filled correctly? e. [1pt] What is the standard deviation of the binomial distribution for the number of orders filled correctly? 2. In a call center, calls arrive at an average rate of 5 calls per minute. (please write calculation process) a. [1pt] What is the probability that exactly 3 calls will arrive in the next minute? b. [ipt] What is the probability that at most 1 call will arrive in the next minute? 3. An industrial sewing machine uses ball bearings that are targeted to have a diameter of 0.85 inch. The lower and upper specification limits under which the ball bearings can operate are 0.84 inch and 0.86 inch, respectively. The experience has indicated that the actual diameter of the ball bearings is approximately normally distributed, with a mean of 0.853 inch and a standard deviation of 0.004 inch. What is the probability that a ball bearing is a. [1pt] between the target and the actual mean? b. [1pt] between the lower specification limit and the target? c. [1pt] above the upper specification limit? d. [1pt] below the lower specification limit? e. [1pt] Of all the ball bearings, 96% of the diameters are greater than what value? 4. Suppose you have a random variable X that follows a uniform distribution on the interval [a, b], where a and b are constants. a. [1pt] What is the probability density function (PDF) of X within the interval [a, b]? Express it mathematically. b. [1pt] Calculate the mean (expected value) of X in terms of a and b. c. [1pt] Determine the variance of X in terms of a and b. d. [1pt] If a = 2 and b = 8, what is the probability that X takes a value between 4 and 6? e. [1pt] If X represents the time (in minutes) it takes for a bus to arrive at a stop, and you know that buses are equally likely to arrive between 8:00 AM and 9:00 AM (a = 480 min, b = 540 min from midnight), what is the probability that the bus will arrive between 8:15 AM and 8:30 AM