1. Consider an experiment where you roll a single six-sided die and observe the number of dots. The sample space is S = {1, 2, 3, 4, 5, 6}. However, the die is unbalanced and the probability each side appears is proportional to its number of dots. For example, on any given roll you are two times more likely to see six dots than you are to see three dots. Find the probability for each outcome in this sample space.

Hint: How does the probability of each other outcome relate to P(1)?

2. A fisherman takes his boat out on a river to catch some trout. Assume that 28% of captured fish are trout, 45% are female, and 12% are female trout. Let T represent the event that a captured fish is a trout, and F represent the event that a captured fish is female. In each part of this question, you must first express each probability in terms of the events T and F and justify any computation using a formula. Do not define any additional events.

(a) Express each of the three probabilities listed above in terms of the events T and F.

(b) Are T and F mutually exclusive events? Explain using the probabilities you stated in part (a).

(c) What proportion of all captured fish are not trout?

(d) What is the probability a captured fish will satisfy at least one of the characteristics of being a trout or being a female?

(e) What proportion of captured fish are female but are not trout?

(f) What is the probability that a captured fish will be neither a trout nor a female?