;;; ========================
;;;  CMPU-145, Spring 2013
;;;  Asmt. 7 Solutions
;;; ========================

(load "asmt7-helper.txt")

(header "Asmt. 7" "Solutions!!")

;;; ============================================================
(problem "1A: Randomly generating a full house:  uniformly!")
;;; ============================================================

;;;  RAND-FULL-HOUSE
;;; ----------------------------------
;;;  INPUTS:  None
;;;  OUTPUT:  A randomly generated hand of 5 cards (i.e., numbers
;;;    from 0 to 51) that has a full house.
;;;  Note:  No particular full house should be any more likely
;;;         than any other!  I.e., uniform probability!!

(define rand-full-house
  (lambda ()
    (let* ((three-ranks (rand-choose-k 3 *ranks*))
           (rank-for-trips (first three-ranks))
           (rank-for-pair (second three-ranks))
           (suits-for-trips (rand-choose-k 3 *suits*))
           (suits-for-pair (rand-choose-k 2 *suits*)))
      (list 
       (card-from-rank-and-suit rank-for-trips (first suits-for-trips))
       (card-from-rank-and-suit rank-for-trips (second suits-for-trips))
       (card-from-rank-and-suit rank-for-trips (third suits-for-trips))
       (card-from-rank-and-suit rank-for-pair (first suits-for-pair))
       (card-from-rank-and-suit rank-for-pair (second suits-for-pair))))))


(tester '(cards->string (rand-full-house)))
(tester '(cards->string (rand-full-house)))
(tester '(cards->string (rand-full-house)))
(tester '(cards->string (rand-full-house)))
(tester '(cards->string (rand-full-house)))
(tester '(cards->string (rand-full-house)))

;;; ============================================================
(problem "1B: EVENT E:  A hand with an even distribution of suits")
;;; ============================================================

;;;  HAS-EVEN-DISTN?
;;; ---------------------------------------------
;;;  INPUT:  CARDS, a list of cards (i.e., numbers from 0 to 51)
;;;  OUTPUT:  #t, if CARDS contains at least one card from each suit.
;;;           #f, otherwise.

(define has-even-distn?
  (lambda (cards)
    (let ((suits (map suit cards)))
      (and (member 0 suits)
           (member 1 suits)
           (member 2 suits)
           (member 3 suits)
           #t))))

(define cards1 (rand-poker-hand))
(tester '(cards->string cards1))
(tester '(has-even-distn? cards1))
(define cards2 (rand-poker-hand))
(tester '(cards->string cards2))
(tester '(has-even-distn? cards2))
(define cards3 (rand-poker-hand))
(tester '(cards->string cards3))
(tester '(has-even-distn? cards3))
(define cards4 (rand-poker-hand))
(tester '(cards->string cards4))
(tester '(has-even-distn? cards4))

;;; ============================================================
(problem "1C:  Some probabilities!!")
;;; ============================================================

;; Number of 5-card hands with even distribution
;;    Note:  One of the suits must appear twice; all others just once.

(define *num-ways-even-distn*
  (* 4  ;; 4 choices for suit that has two
     (n-choose-k 13 2)  ;; 2 ranks for that suit
     13 ;; 13 choices for rank for each of the remaining suits
     13
     13))

;; Probability of getting dealt a hand with even distribution

(define *prob-even-distn*
  (/ *num-ways-even-distn*
     *num-5-card-hands*
     1.0))

(tester '*prob-even-distn*)

;; Number of hands with full houses

(define *num-full-houses* 
  (* 13 ;; choices for rank of trips
     12 ;; choices for rank of pair
     (n-choose-k 4 3) ;; suits for trips
     (n-choose-k 4 2) ;; suits for pair
     ))

;;  Probability of getting dealt a full house

(define *prob-full-house*
  (/ *num-full-houses*
     *num-5-card-hands*
     1.0))

;;  Probability of a hand having even distribution given that
;;  it is a full house.

(define *prob-even-distn-given-full-house*
  .50 
  ;; The trips in the full house must use three different suits:  call 
  ;; them A, B, and C.  So we only need to know the probability that ONE
  ;; of the pair will have the remaining suit, call it D.
  ;; Since the suits of the pair have to be different, there are 4-choose-2
  ;; ways of picking the suits for the pair:  (A B), (A C), (A D),
  ;; (B C), (B D), (C D).  Notice that half of them involve the
  ;; fourth suit, D.
  )

;; Probability of a hand having even distribution AND being a full house!
;;   Note:  We use:  P(E AND F) = P(E|F)*P(F)

(define *prob-even-distn-and-full-house*
  (* *prob-even-distn-given-full-house*
     *prob-full-house*))

;; Probability of a hand being a full house given that it has even distn.
;;   Note:  We use:  P(F|E) = P(F AND E) / P(E) = P(E AND F) / P(E)

(define *prob-full-house-given-even-distn*
  (/ *prob-even-distn-and-full-house*
     *prob-even-distn*))

;; SHow me!

(tester '*prob-even-distn-given-full-house*)
(tester '*prob-full-house*)
(tester '*prob-full-house-given-even-distn*)
(tester '*prob-even-distn-and-full-house*)
(tester '(* *prob-even-distn* *prob-full-house*))
(printf "Note that the events are NOT independent, since the last two values are different.~%")

;;; ============================================================
(problem "1D: Using Gen-And-Test")
;;; ============================================================

;;; This should estimate Prob(Even Distn|Full House)

(tester '(gen-and-test 100000 rand-full-house has-even-distn?))

;;; ============================================================
(problem "1E: Generating random hands with even distribution")
;;; ============================================================

;;;  RAND-EVEN-DISTN
;;; ---------------------------
;;;  INPUT:  none
;;;  OUTPUT:  A randomly generated hand that has at least one
;;;           card in each suit.
;;;  NOTE:  Each hand with even distribution should be equally likely.

(define rand-even-distn
  (lambda ()
    (let* (;; Pick the suit that will appear twice
           (twofer-suit (rand-choose-one *suits*))
           ;; The remaining suits will each appear once
           (other-suits (remove twofer-suit *suits*))
           ;; Pick two ranks for the TWOFER_SUIT
           (ranks-for-twofer (rand-choose-k 2 *ranks*)))
      ;; Construct the list of five cards
      (list (card-from-rank-and-suit (first ranks-for-twofer) twofer-suit)
            (card-from-rank-and-suit (second ranks-for-twofer) twofer-suit)
            (card-from-rank-and-suit (rand-choose-one *ranks*) 
                                     (first other-suits))
            (card-from-rank-and-suit (rand-choose-one *ranks*)
                                     (second other-suits))
            (card-from-rank-and-suit (rand-choose-one *ranks*)
                                     (third other-suits))))))

(tester '(cards->string (rand-even-distn)))
(tester '(cards->string (rand-even-distn)))
(tester '(cards->string (rand-even-distn)))
           
;;; ============================================================
(problem "1F: And using GEN-AND-TEST one more time!")
;;; ============================================================

;;;  This should estimate Prob(FullHouse|Even Distn)

(tester '(gen-and-test 1000000 rand-even-distn has-full-house?))