The following questions assess your knowledge of identifying the sample space of a chance experiment, calculating probabilities using equally likely outcome model (the f/N rule) if applicable, calculating probabilities of events using addition, complementation, conditional, multiplication rules, showing whether two events are independent by calculation, and using combination and permutation to calculate the number of sample points of events.

1. Let H be the event of observing ahead and T be the event of observing a tail. A balanced coin is tossed three times.

(a) List all possible outcomes.

(b) List all possible outcomes and find the probabilities of the following events.

A = event exactly two heads are tossed

B = event the first toss is a tail

C = event the first toss is a head

D = event all three tosses come up the same

(c) List all possible outcomes and find the probabilities of the following events.

(1) not A (2) A & B (3) B & C (4)C or D (5) D|A

(d) Identify all possible pairs of events defined in part (b) that are mutually exclusive.

(e) Are the events A and D independent? Explain your answer mathematically.