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Theory (100 points) – please show your work!! 1. (30 points/5 points each) Suppose two dice are tossed (for each die, it is equally likely that 1, 2, 3, 4, 5, or 6 dots will show). a. What is the probability that the total of the two dice will add up to 7 or 11? b. What is the probability that the total of the two dice will add up to a number other than 2 or 12? c. Are the following events independent? i. E1 = first die shows a 3. ii. E2 = total of the two dice is 6. d. Are the following events independent? i. E1 = first die shows a 3. ii. E2 = total of the two dice is 7. e. Given that the total of the two dice is 5, what is the probability that the first die showed 2 dots? f. Given that the first die shows 5, what is the probability that the total of the two dice is even? 2. (20 points) A desk contains three drawers. Drawer 1 contains two gold coins. Drawer 2 contains one gold coin and one silver coin. Drawer 3 contains two silver coins. I randomly choose a drawer and then randomly choose a coin. If a silver coin is chosen, what is the probability that I chose drawer 3? 3. (20 points) Joon wants to determine whether his San Jose oil field will yield oil. He has hired geologist to run tests on the field. If there is oil in the field, there is 95% chance that the test will indicate oil. If the field contains no oil, there is a 5% chance that the test will indicate oil. If the test indicates that there is no oil in the field, what is the probability that the field contains oil? Before the test, Joon believes that there is a 10% chance that the field will yield oil. 4. (30 points/ 5 points each) Farmer Joon must determine whether to plant corn or wheat. If the plants corn and the weather is warm, he earns $8,000; if he plants corn and the weather is cold, he earns $5,000. If he plants wheat and the weather is warm, he earns $7,000; if he plants wheat and the weather is cold, he earns $6,500. In the past, 40% of all years have been cold and 60% have been warm. Before planting, Joon can pay $600 for an expert weather forecast. If the year is actually a cold, there is a 90% chance that the forecaster will predict a cold year. If the year is actually warm, there is an 80% chance that the forecaster will predict a warm year. a. What is the probability that the year will be actually warm if expert said “will be warm”? b. What is the probability that the year will be actually warm if expert said “will be cold”? c. What is the probability that the year will be actually cold if expert said “will be cold”? d. What is the probability that the year will be actually cold if expert said “will be warm”? e. If expert said “will be cold”, then which crop do we plant this year?