9/20/96 - 1001, 2001

Write your answers in the space provided. You are encouraged to use calculators.

1. Two different (but similar) screening tests are given to plebes. 30% of the plebes test below standard on Test 1 and 25% test below standard on Test 2. Further, 12% test below standard on both tests.
   1. Write the Venn diagram for this problem.
   2. How many students score above standard on both tests?
   3. If a student tests below standard on Test 1, what is the chance he will test below standard on Test 2?
   4. If a student tests below standard on Test 2, what is the chance he will test below standard on Test 1?
   5. Is the 12% figure more or less than expected if the two tests were independent?
2. The orange army is 30% regular and 70% volunteer. Only 10% of the regular army is expected to surrender, while 50% of the volunteers will.
   1. What fraction of the whole army is expected to surrender?
   2. Among those who surrender, how many are regular army?
   3. Is surrendering independent of regular/volunteer? Why?
3. The cowboy has to find the lost dogie (motherless calf). He figures it is either in the ravine, in the barn or on the next ranch. Being rather enlightened for someone who needs to wear boots on his job, he figures the probabilities of the dogie being in each location are 0.5 (ravine), 0.35 (barn), 0.15 (next ranch). The probabilities of being detected if the location is searched are 0.20, 0.80 0.30.
   1. Write the Venn diagram for searching all 3 locations.
   2. If he searches all 3 locations, what is the probability of finding the dogie?
   3. If the dogie is not found, what is the probability it is in the barn?
   4. If he searches all 3 locations a second time (not finding it the first time) and fails again to find the dogie, what is the probability it is in the ravine?
   5. If he had started out by searching only the barn, instead of all 3 locations, and did not find the dogie, what is the chance it is on the next ranch?
4. "Let's Make a Deal!" We will play using 4 doors. Monty will not show us our door, nor the door with the good prize. When he has a choice of doors to show us, he will choose the doors with equal probability. (If 3 doors are possible, each has probability 1/3. If 2 doors are possible, they each have probability 1/2.) We will start by choosing Door #1. Monty then shows us a bad prize behind Door #2. The twist is that Door #1 has probability 0.4 of having the prize. Door #2 has probability 0.15, Door #3 has probability 0.20 and Door 4 has probability 0.25.
   1. Find Prob(Prize behind #1 | Monty shows #2)
   2. Find Prob(Prize behind #3 | Monty shows #2)
   3. Find Prob(Prize behind #4 | Monty shows #2)
   4. Now which door should we pick (having seen Door #2)?
5. Suppose that driving a foreign car is independent of whether a faculty member is military or civilian. If 45% of the faculty is military and 65% of the faculty drive foreign cars, what fraction are civilian and drive domestic cars?