353 - Topic 03 - Conditional/Independent -- Sage

John Travis

Mississippi College

MATH 353 - Introduction to Mathematical Probability and Statistics

Textbook: Tanis and Hogg, A Brief Course in Mathematical Statistics

Conditional vs independent events

%auto var('Ace Clubs Diamonds Hearts Jack King Queen Spades') suits = [Spades, Diamonds, Clubs, Hearts] values = [2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace] deck = [(value, suit) for suit in suits for value in values] full_deck = copy(deck) # to save a copy of the original deck for later use. pretty_print(html('Consider the full standard deck of 52 cards as a collection of ordered pairs (face,suit)')) for value in values: show([(value,suit) for suit in suits])

Consider the full standard deck of 52 cards as a collection of ordered pairs (face,suit)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(2, \mathit{Spades}\right), \left(2, \mathit{Diamonds}\right), \left(2, \mathit{Clubs}\right), \left(2, \mathit{Hearts}\right)\right]$

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(3, \mathit{Spades}\right), \left(3, \mathit{Diamonds}\right), \left(3, \mathit{Clubs}\right), \left(3, \mathit{Hearts}\right)\right]$

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(4, \mathit{Spades}\right), \left(4, \mathit{Diamonds}\right), \left(4, \mathit{Clubs}\right), \left(4, \mathit{Hearts}\right)\right]$

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(5, \mathit{Spades}\right), \left(5, \mathit{Diamonds}\right), \left(5, \mathit{Clubs}\right), \left(5, \mathit{Hearts}\right)\right]$

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(6, \mathit{Spades}\right), \left(6, \mathit{Diamonds}\right), \left(6, \mathit{Clubs}\right), \left(6, \mathit{Hearts}\right)\right]$

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(7, \mathit{Spades}\right), \left(7, \mathit{Diamonds}\right), \left(7, \mathit{Clubs}\right), \left(7, \mathit{Hearts}\right)\right]$

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(8, \mathit{Spades}\right), \left(8, \mathit{Diamonds}\right), \left(8, \mathit{Clubs}\right), \left(8, \mathit{Hearts}\right)\right]$

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(9, \mathit{Spades}\right), \left(9, \mathit{Diamonds}\right), \left(9, \mathit{Clubs}\right), \left(9, \mathit{Hearts}\right)\right]$

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(10, \mathit{Spades}\right), \left(10, \mathit{Diamonds}\right), \left(10, \mathit{Clubs}\right), \left(10, \mathit{Hearts}\right)\right]$

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(\mathit{Jack}, \mathit{Spades}\right), \left(\mathit{Jack}, \mathit{Diamonds}\right), \left(\mathit{Jack}, \mathit{Clubs}\right), \left(\mathit{Jack}, \mathit{Hearts}\right)\right]$

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(\mathit{Queen}, \mathit{Spades}\right), \left(\mathit{Queen}, \mathit{Diamonds}\right), \left(\mathit{Queen}, \mathit{Clubs}\right), \left(\mathit{Queen}, \mathit{Hearts}\right)\right]$

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(\mathit{King}, \mathit{Spades}\right), \left(\mathit{King}, \mathit{Diamonds}\right), \left(\mathit{King}, \mathit{Clubs}\right), \left(\mathit{King}, \mathit{Hearts}\right)\right]$

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(\mathit{Ace}, \mathit{Spades}\right), \left(\mathit{Ace}, \mathit{Diamonds}\right), \left(\mathit{Ace}, \mathit{Clubs}\right), \left(\mathit{Ace}, \mathit{Hearts}\right)\right]$

Consider the full standard deck of 52 cards as a collection of ordered pairs (face,suit)

Let's consider the difference between picking two cards with replacement vs picking two cards without replacement.

%auto print "Conditional Events - successively deal 5 cards w/o replacement" deck1 = copy(full_deck) history1=[] @interact def _(choice=['Hearts','Spades','Diamonds','Clubs','New Deck']): global deck1, history1 shuffle(deck1) if choice=='Hearts': suit = Hearts elif choice=='Spades': suit = Spades elif choice=='Diamonds': suit = Diamonds elif choice=='Clubs': suit = Clubs else: deck1 = copy(full_deck) shuffle(deck1) history1=[] if (Set(deck1).cardinality()<5): pretty_print('Deck is too small...get a new deck') elif choice<>'New Deck': hand = [deck1.pop() for card in range(5)] print "Click on a desired suit above to deal out another 5 card hand. The cards dealt:" show(hand) print "The remaining cards in the deck:" print deck1 num = Set(deck1).cardinality() pretty_print("\nThe number of remaining cards in the deck = %s"%str(num)) looking = [] for card in deck1: if card[1]==suit: looking.append(card) prob = float(Set(looking).cardinality())/num history1.append(prob) pretty_print('So, the remaining probability of getting a card from '+choice+' from the remaining cards is %s'%str(prob)) list_plot(history1).show(xmin=0,xmax=9,ymin=0,ymax=1,figsize=(5,2))

Conditional Events - successively deal 5 cards w/o replacement

choice

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Notice that the probability changes of getting a card from a given suit changes as progressive hands are dealt if the cards are not replaced before redealing.

%auto print "Independent Events - Successively deal 5 cards but WITH replacement" #deck1 = copy(full_deck) history2=[] @interact def _(choice=['Heart','Spade','Diamond','Club']): if choice=='Hearts': suit = Hearts elif choice=='Spades': suit = Spades elif choice=='Diamonds': suit = Diamonds else: suit = Clubs deck1 = copy(full_deck) shuffle(deck1) hand = [deck1.pop() for card in range(5)] print "The cards dealt:" show(hand) print "Replacing this hand and reshuffling gives the remaining cards in the deck:" deck1 = copy(full_deck) shuffle(deck1) print(deck1) num = Set(deck1).cardinality() pretty_print("\nThe number of remaining cards in the deck = %s"%str(num)) looking = [] for card in deck1: if card[1]==suit: looking.append(card) prob = float(Set(looking).cardinality())/num history2.append(prob) pretty_print('So, the remaining probability of getting a '+choice+' from the remaining cards is %s'%str(prob)) list_plot(history2).show(xmin=0,xmax=9,ymin=0,ymax=1,figsize=(5,2))

Independent Events - Successively deal 5 cards but WITH replacement

choice

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The stuff below is just playing around with sampling. Don't bother with this right now.

Events are independent provided the occurance of a first event does not likelihood of a subsequent event.

Create an interact which picks random integers from 1 to 10 and then picks another random number from 1 to 10.

Two cases:

the first selection is removed and the second selection can come from only the remaining integers
the first selection is recorded and then "replaced" so that the second selection can come from any of the original integers

@interact def _(n = slider(10,2000,30,10,label='Number of samples')): Samples = range(n) max_pt=20 Domain = range(max_pt) A = [ZZ(int(max_pt*(random()))) for i in Samples] A1 = IndexedSequence(A,Samples) F = frequency_distribution(A) show(F) # G = points((x,F(x)) for x in Domain) G = A1.plot_histogram(eps=0.45,clr='green') G.show(ymin=0,ymax = max_pt,figsize=(10,3))

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low = 0 high = 10 @interact def _(n=slider(1,100,1,label='Number of Trials'),replace=checkbox(false,label='With replacement?')): A = random_matrix(ZZ, 1, n, x=1, y=high+1) A1= zero_matrix(1,high) print A1 for k in range(1,high+1): print 'hi' temp = A(k) A1(temp) = A1(temp)+1 print A1 R = [low..high] A = IndexedSequence(A,R) G=A.plot_histogram(eps=0.5,clr='green') show(G,ymin=0)

353 - Topic 03 - Conditional/Independent

2403 days ago by Professor353

Click to the left again to hide and once more to show the dynamic interactive window

Click to the left again to hide and once more to show the dynamic interactive window

Click to the left again to hide and once more to show the dynamic interactive window

Click to the left again to hide and once more to show the dynamic interactive window