# Practice Exam for Second Midterm

### Directions:

• Answer the problems on the exam pages.
• There are seven problems on pages 2-7, for 100 total points. Probable scale is A=93, C=63.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• The first five questions are true/false, with five points for the correct boolean answer and up to five for a correct justification.
• Question 6 has numerical answers -- you may give your answer in the form of an expression using arithmetic operations, powers, falling powers, or the factorial function. If you give your answer using the "choose" notation, also give it using only operations on this list.

```  Q1: 10 points
Q2: 10 points
Q3: 10 points
Q4: 10 points
Q5: 10 points
Q6: 30 points
Q7: 20 points
Total: 100 points
```

• Question 1 (10): True or false with justification: Suppose I make four successive independent throws of a single six-sided die. Then the probability that I ever throw the same number twice in a row is less than or equal to 1/2.

• Question 2 (10): True or false with justification: Let A, B, and C be three pairwise independent random variables over the same event space. (This means that A and B, A and C, and B and C are each independent pairs.) Then Prob(A|B) = Prob(A|C).

• Question 3 (10): True or false with justification: If I collect 2n coupons, each of which is independently and uniformly chosen from one of n types, the probability that I get at least one coupon of each type is no more than 1 - 1/e2.

• Question 4 (10): True or false with justification: It is not possible for a random variable to always take on a value that is at least one standard deviation away from its mean.

• Question 5 (10): True or false with justification: Let X be a variable that always takes on a value less than -1, and whose mean is -2. It is not possible that Pr(X ≤ -5) = 10%.

• Question 6 (30): In college basketball, there are two different ways a player can be awarded two free throws. In a one-and-one, the player attempts her first throw and is allowed a second throw only if the first throw succeeds. In a two-shot foul, the player is allowed the second throw whether the first is successful of not. Assume that Julia's free throws are independent events, each with a 2/3 probability of success.

• (a,5) If Julia shoots a one-and-one, what is the probability that she gets two points? That she gets one point? That she gets no points? (Each successful free throw counts for one point.)

• (b,5) What is her expected number of points in a one-and-one?

• (c,5) What is the variance of the number of points she gets in a one-and-one? (You may find the formula Var(X) = E(X2) - E(X)2 to be useful.)

• (d,5) What are the probabilities of 0, 1, or two points respectively if Julia shoots a two-shot foul?

• (e,5) What is her expected number of points for a two-shot foul?

• (f,5) What is the variance of the number of points she gets in a two-shot foul? (Hint: This question is substantially easier than part (c) if you see how to do it.)

• Question 7 (20): Recall that B(n,p,i) is the probability that there are exactly i successes in a sequence of n Bernoulli trials where each trial has success probability p. Prove that for any natural number k, B(2k,0.5,k) ≥ 2-k. (Hint: Use induction on k and argue that if you get k successes in the first 2k trials, there is at least a 50% chance that you will get k+1 successes in the first 2k+2 trials.)