**2. **(**12 points**) A statistical experiment involves flipping a fair coin and rolling a fair, six-sided die. Due to a manufacturing error, two sides of the fair die show 4; it has no 5 on it; other four sides show 1, 2, 3, or 6.

(a) Determine the sample space for the statistical experiment. The sample space is the set of all possible outcomes; for example, 3H, 5T, where the single digit comes from the die and H or T comes from the coin, H standing for Heads and T standing for Tails. The format of your sample space (S) in set notation would look like

S = {1H, 1T, … }

**Hint:** You may use a tree diagram or a table to find all possible outcomes in an organized manner to catch all possible outcomes.

(b) If H represents 1 and T represents 2, construct a two-column __probability distribution table__ for the random variable X, representing the total for each possible outcome and its probability (a fraction). For example, for 3H, the value of X would be 3+1 = 4, and for 6T the value of X

would be 6+2 =8.

**Hint:** Make sure that the probability values (preferably in fractions) in the probability column of your table add up to 1.

(c) Using your table, find the expected value of X, as E(X) = ?( xi P(xi) ).