# Midterm Exam

### Directions:

• Answer the problems on the exam pages.
• There are six problems for 100 total points. Probable scale is A=90, C=63.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• The first four questions are true/false, with five points for the correct boolean answer and up to five for a correct justification.
• The other questions have 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: 20 points
Q6: 40 points
Total: 100 points
```

The four true/false questions all involve a situation where two cards are drawn, without replacement, from a standard 52-card deck, such that each pair of cards is equally likely.

• Question 1 (10): True or false with justification: The probability that the two cards have the same rank (that is, that they form a pair) is exactly 1/17.

• Question 2 (10): True or false with justification: The chance that the hand contains one or more aces is exactly [(4 choose 2) + 4(48)]/(52 choose 2).

• Question 3 (10): True or false with justification: The chance that the two cards form a flush (have the same suit) is less than the chance that they form a straight (have two adjacent ranks, i.e., A-2, 2-3, 3-4,...,Q-K, K-A). (For purposes of this question, consider a straight flush to be both a straight and a flush.)

• Question 4 (10): True or false with justification: The chance that the hand contains either one or more aces, one or more kings, or both is more than twice the probability that it contains one or more aces.

• Question 5 (20): Consider a game where we toss a fair coin repeatedly until we get the same result twice in a row. Let the random variable X be the total number of coin tosses including the last two that are the same -- for example, if the first two tosses are both heads then X = 2.

• (a,5) Find the probability that X = 1, the probability that X = 2, the probability that X = 3, and the probability that X = 4.

• (b,5) For an arbitrary positive integer n, find the probability that X = n, as a function of n. Explain your answer.

• (c,10) Compute E(X), the expected value of X.

• Question 6 (40): Every U.S. dollar bill has a serial number, which has eight digits. Assume that for a random dollar bill, each of these digits is equally likely to be any number in the set {0,1,2,3,4,5,6,7,8,9}, and that the digits are independent of each other.

• (a,5) Compute the probability that the eight digits on a random dollar bill are all different.

• (b,5) How many possibiltities are there for the multiset of digits occurring on a dollar bill? (One example of such a multiset is "three 4's, four 8's, and on 9".)

• (c,5) How many possibilities are there for the set of digits occurring on a dollar bill? (In the example above, the set would be {3,8,9}.)

• (d,5) Compute the probability that the eight digits on a random dollar bill divide into four copies of one digit and four of another digit. (I want the total for all choices of the two digits.)

• (e,10) Compute the probability that a random dollar bill has exactly four copies of some digit. (Hint: This could happen in several ways -- the situation in (d), which we could call 4-4, but also 4-3-1, 4-2-2, 4-2-1-1, or 4-1-1-1. You might also be able to use Inclusion/Exclusion.)

• (f,10) Let F be a random variable equal to the number of 4's on a random dollar bill. Compute the expected value of F. Compute the variance of F.