;; The first three lines of this file were inserted by DrRacket. They record metadata
;; about the language level of this file in a form that our tools can easily process.
#reader(lib "htdp-intermediate-lambda-reader.ss" "lang")((modname 22prob) (read-case-sensitive #t) (teachpacks ((lib "image.rkt" "teachpack" "2htdp") (lib "universe.rkt" "teachpack" "2htdp"))) (htdp-settings #(#t constructor repeating-decimal #f #t none #f ((lib "image.rkt" "teachpack" "2htdp") (lib "universe.rkt" "teachpack" "2htdp")))))
;; DEMO: probability
#|
DATA DEFINITION

A (weighted X) is a structure

  - (make-weighted weight value)
    where `weight` is a number and `value` is an X

A (dist X), aka "probability distribution of X" is
a (listof (weighted X)) where the weights in the list add up to 1
|#
(define-struct weighted (weight value))

;; payoff : natural -> number
;; give the payoff in dollars for the given roll in the "straight-line" dice game
(define (payoff roll)
  (cond [(= 0 roll) 0]
        [(= 1 roll) 1]
        [(= 2 roll) 1]
        [(= 3 roll) 0]
        [(= 4 roll) 3]
        [(= 5 roll) 2]
        [(= 6 roll) 0]
        [(= 7 roll) 2]
        [(= 8 roll) 0]
        [(= 9 roll) 0]
        [else (+ (payoff (remainder roll 10))
                 (payoff (quotient roll 10)))]))

(check-expect (payoff 7) 2)
(check-expect (payoff 18) 1)
(check-expect (payoff 14) 4)


;; d : number -> (dist number)
;; compute the probability distribution of rolls for a die with N sides
(define (d n)
  (build-list n (lambda (i) (make-weighted (/ 1 n) (add1 i)))))

(check-expect (d 4) (list (make-weighted 1/4 1)
                          (make-weighted 1/4 2)
                          (make-weighted 1/4 3)
                          (make-weighted 1/4 4)))

;; expectation : (dist X) (X -> number) -> number
;; compute the average result from applying `f` to values drawn from `dist`
(define (expectation dist f)
  (foldl
   (lambda (wv sum) (+ sum (* (weighted-weight wv) (f (weighted-value wv)))))
   0
   dist))

(check-expect (expectation (d 4) payoff)
              5/4)
          

;; probability : (dist X) (X -> boolean) -> number
;; compute the probability of `(p? x)` when `x` drawn randomly from `dist`
(define (probability dist p?)
  ...)

;(check-expect (probability (d 6) (lambda (n) (< n 5)))
;              ...)



;; Odds of dice 1 vs dice 2: P(observation|dice1) / P(observation|dice2)

;; How much *evidence* is added by an observation?