Midterm #2, Spring 2022 Page 7 of 14 Taaffe ISE 2024 6. (9 points) A package of contains 50 low-wattage light bulbs and 50 high-wattage light bulbs. (a) (3 points) If a random sample of ten light bulbs is taken without replacement from the box, what is the numerically exact probability that at least four of each type of light bulb is in the sample? (b) (3 points) As an approximation to the calculation from part (a), assume that the random sampling is performed with replacement. What is the exact probability that at least four of each type of light bulb is in a random sample drawn with replacement from the box? (c) (3 points) Now using the Poisson approximation to the binomial distribution, approx- imate the solution to part (b). Comment on why this approximation is either good or not so good. Report all numbers to 4 digits. Let X be the number of low-wattage bulbs in the sample. Hint: . Think about what kind of distribution X has when there is replacement. . Think about what kind of distribution X has when there is no replacement. . Since X is the number of low-wattage light bulbs in the sample of size 10, then it must be that the number of high-wattage bulbs in the sample is simply 10 - X. Thus only one computation about X tells also tells us about 10 - X