# 1. The Scheme-- programming language

The Scheme-- programming language is a simplified version of lambda calculus that incorporates the exciting reference (`&`) and dereference (`*`) operators from C but otherwise doesn't actually work. Scheme-- expressions are of the form

<> (the empty string)

`x``y``17``&`α`*`α`λx`α`λy`α`(`α`)`

where in each case α represents another Scheme-- expression. The lack of variables other than `x` and `y`, constants other than `17`, and any notion of function application, is a feature of the language, intended to encourage users to write simple (though useless) programs. Some examples of Scheme-- expressions: `&*λxy`, `(**&λyλy(17))`, `x`. These have length 5, 13, and 1, respectively.

Compute the number of Scheme-- expressions of length n.

# 2. Independent events

Let A and B_{1} be independent events, and let A and B_{2} also be independent events, on some probability space Ω.

Prove or disprove: A and Ω-B

_{1}are independent events.Prove or disprove: A and B

_{1}∩B_{2}are independent events.Prove or disprove: If A and B

_{1}∪B_{2}are independent events, then B_{1}and B_{2}are independent events.

# 3. Small poker hands

A standard 52-card poker deck contains one each of the cards {A,2,3,4,5,6,7,8,9,10,J,Q,K} in each of the four suits {♣,♢,♡,♠}. The J (Jack), Q (Queen), and K (King) cards collectively make up the 12 face cards.

Suppose the deck is shuffled uniformly (so that all 52! orderings are equally likely), and you are dealt the first two cards.

- What is the probability that both cards are face cards?
- What is the probability that both cards are face cards, given that the first card you are dealt is a Jack?
- What is the probability that both cards are face cards, given that exactly one of them is a Jack?
- What is the probability that both cards are face cards, given that at least one of them is a Jack?