[20 pts total] Solve the following problems. For this question you must derive probability mass functions from first principles (e.g. using counting rules and properties of probabilities) instead of attempting to use the "named" distributions that we will discuss next week. Note: when writing down probability mass functions, the range of the variable must always be clearly provided.

(a) [5 pts] Tay-Sachs disease is a rare but fatal disease of genetic origin occurring mostly in infants and children, especially those of Jewish or eastern European descent. If a couple are both carriers of Tay-Sachs disease, a child of theirs has probability .25 of being born with the disease. If such a couple has four children, what is the probability mass function for the number of children who will have the disease? What important assumption did you make? Please define all notation, state any assumptions, and clearly state the probability mass function as a formula.

(b) [5 pts] My spam filter flags 2% of my( emails), which I assume arrive in my inbox at random. Let the random variable Y be the number of emails I receive until I see the second spam message (including the second spam message). Write down the probability mass function of Y as a formula. [Hint: write down p(y) for a few cases and then try to guess a formula for a general y from the pattern]

(c) [5 pts] A jar contains 5 blue and 10 green marbles. You draw 3 marbles out of the jar without replacement. Let Y be the number of the drawn marbles that are blue. Write down the probability mass function of Y as a formula.

(d) [5 pts] Suppose that a random variable Y takes only the following values: y= 1,2,3,4. The probability mass function for the three values y = {1, 2, 3} is p(y) = (y2 + 4y 1)/40, y = 1, 2, 3. However, the probability that Y = 4 is not provided. Find P(Y = 4).