Question: Consider what the weather might be on October 20 of next year. If we are considering planning an outdoor birthday party for this day, we may think that two distinctions about the weather are important: whether or not it rains, and what the temperature is. We may even believe that they are relevant to each other
a. Let R designate the event that it rains. Formulate the distinctions R and R' (where R' means "Not R") so that they are mutually exclusive and collectively exhaustive. Subject your distinctions to the clarity test. Let W represent that the same day is warm and W represent not warm. Formulate this distinction clearly. Without attaching any numbers, draw a tree and label all the possibilities created by these two distinctions
b. Let & stand for your background state of information. In words define the following: {R | &), {VF| /?, &), {R| W,&}, {R,W|&}
c. Discuss the relevance between R and W. Are they both caused by the same weather systems? Does a warm day cause rain? Does rain cool down the temperature?
d. Fill in the probabilities on a tree with R and R' on the first set of branches and with W and W on the second set. Flip the tree and note the resultant probabilities that this assessment produces for W and W. Adjust the conditional probabilities until you are happy with your overall assessment. Note that flipping the tree can serve as a check on your initial assessments. Iterating a few times will usually produce an assessment with which you feel comfortable.