1. [10 points] A software company is developing three applications: mobile, web, and desktop. Let B,, Bo, and Bs denote the events that each is completed on time. Using union, intersection, and complement, express the following events in terms of B,, Bj, and B3;. Draw a Venn diagram and shade the corresponding region. a. All apps are completed on time. b. Only the mobile app is completed on time. c. Exactly one app is completed on time. 2. [10 points] At a coffee shop, 45% of customers request whipped cream (D), 60% caramel drizzle (), and 75% an espresso shot (f'). Also, 68% request D or E, 82% request D or F,, 86% request EH or F', and 90% request at least one of the three add-ons. Determine the probabilities of the following events. a. The next customer requests at least one add-on. b. The next customer requests none of the add-ons. c. The next customer requests exactly one of the three add-ons. 3. [10 points] A box of 25 smartphones is being examined for defects. Of these, 6 have screen defects and 19 have battery defects. a. How many ways can 5 smartphones be randomly selected for inspection (order does not matter)? b. In how many ways can 5 smartphones be selected such that exactly 2 have screen defects? c. If 5 smartphones are randomly selected, what is the probability that at least 4 have battery defects? 4. [10 points] Show that for any three events A, B, and C with P(C) > 0, the following holds: P((AUB)|C) = P(A|C)+ P(B|C) P(ANB|C). 5. [10 points] An individual from a population has a 5% chance of having Type II diabetes. A diagnostic test detects the disease correctly 98% of the time (true positive rate) and correctly identifies non-diabetics 99% of the time (true negative rate). The test is applied twice independently. If both results are positive, what is the probability that the individual actually has Type II diabetes