Homework #5 on Probability Trees

Instructions: Solve using the conditional probabilities for the exercises by constructing probability-tree diagrams. Do the exercises in either fractions or decimals as directed in each exercise. For each exercise, do the probability tree first, and then state your answer(s) below it. (I give you the conditional probability expressions, e.g. Pr(A|B), which means, "Probability A if B," just to get you used to seeing them. They are otherwise irrelevant to this homework assignment. In other words, you do not need to worry about them for this homework. That said, if you are really on the ball, they may help you figure out how many branches you need.)

Exercise #1 (use decimals)... : Of the 100 members in the United States Senate, there are 55 republicans and 45 democrats. There is a bill up for vote. 20% of republicans, Pr(S|R), and 80% democrats, Pr(S|D), support it. You are at the Capitol Building for a hearing related to the bill. All the senators are there. Using a probability tree, figure out what the probability is of your bumping into 1) a republican who supports the bill, 2) a democrat who does NOT support the bill, and 3) any senator who supports the bill. Express your final answer as percentages.

Exercise #2 (use fractions)... : A box contains three coins: one coin is fair; one is two-headed, and one is weighted so that the probability of flipping tails is two-thirds and heads is one-third. You randomly select one coin, meaning that each coin has a one-third chance of being selected. Using a probability tree, figure out what the probability is of your flipping heads, Pr(H|F) or Pr(H|T) or Pr(H|O).

Exercise #3 (use decimals)... : 90% of the local meteorologists predicted rain for today. 10% did not. Studies indicate that meteorologists are right 70% of the time, Pr(R|P) and Pr(R|~P). Those meteorologists who predicted rain for today forecast an 80% chance of rain, Pr(F|P&R). Using a probability tree, figure out 1) what the probability is that the meteorologists who predicted rain will end up being right and 2) what the chances of rain are (that is, those who predicted rain end up being right and those who did not, end up being wrong). Express your final answers as percentages.

Exercise #4 (use fractions)... : An automobile parts store, Good Carma, sells two brands of shock absorbers. They get three-fifths of their stock of shock absorbers from Motor Parts, Inc., and two-fifths from Acme & Co. Five out of six of the shock absorbers Good Carma gets from Motor Parts, Inc., exceed the industry standard in reliability, Pr(E|M). Three out of four of the shock absorbers they get from Acme & Co. exceed the industry standard in reliability, Pr(E|A). Using a probability tree, figure out what the chances are of purchasing a random shock absorber at Good Carma that exceeds the industry standard in reliability. Express your final answer in a fully reduced fraction.