Part IV: Probability

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Part IV Probability Chapters 13 to 15
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Chapter 13 Probability what are the chances???

• The word chance is used often
• what are the chances that we win the national
  championship?
• what are my chances of getting an A in class??
• if the population of undergraduates here is
  52 male, what proportion of a random sample of
  100 drawn from it will be male - close to
  52 how close is close?
• Chance needs a clear definite mathematical
  interpretation
• provided by frequency theory
• works best for processes repeated over and
  over, independently, under the same conditions
• example games of chance (gambling)
• Example tossing a coin and betting
  which side turns up the process can be
  replicated, independently under the same
  conditions chances of getting a head is 50 - in
  the long run, heads will turn up about 50 of
  the time

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Chapter 13 Probability what are the chances???
Example rolling a die
a cube with six faces labeled one through
six when the die is rolled, the faces are
equally likely to turn up the chance of getting
a 6 1/6 or 16 2/3 thus if the process of
rolling a fair die were to be repeated over and
over under the same conditions, over the long
run a 6 will show up 16 2/3 of the time
Chance the of time something IS EXPECTED to
happen when the basic process is repeated over
and over, independently and under the same
conditions
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Chapter 13 Probability what are the chances???
Example rolling a die
rolling a die and getting 10 is impossible - it
happens 0 of the time rolling a die and getting
a non-blank face happens 100 of the time all
chances fall between 0 and 100 if there are two
outcomes only (example a coin toss, winning or
losing), the chance of one outcome 100 minus
the chance of the opposite outcome
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A 20 fresh 10 sour
B6 fresh 3 sour
2 possible outcomes fresh or sour milk
Which offers the best chance of fresh milk?
Drawing from either over the long run with
replacement produces a
2 in 3 chance of fresh milk for both refrigerators
Note drawing at random all items/people
etc have the SAME chance to be picked
chance/probability number of fresh
cartons total number of cartons
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Chapter 13 Box Models

• Many problems can be conceptualized (like the
  previous example) as random draws from a box
• typical instructionsdraw 2 tickets at random
  WITH replacement from the box

1
2
3
Process shake the box draw one ticket out at
random (all 3 have the same chance of being
selected) note the number, say the ticket
numbered 3, and PLACE it back in the box repeat
the process again
Note that for both draws, the probability of
drawing any number remains the same for both
draws!! i.e. one out of 3
1
2
3
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Chapter 13 Box Models
Compare this to drawing 2 tickets at random
WITHOUT replacement from the box
1
2
3
Process shake the box draw one ticket out at
random (all 3 have the same chance of being
selected) note the number, say the ticket
numbered 3, and and DO NOT PLACE it back in the
box repeat the drawing
Note that the probability of drawing a number on
the second draw changes as the box now becomes
1
2
When you draw at random, all tickets have the
same chance of being selected
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Chapter 13 Exercise Sets 1. Someone computes
the chances for various events. Match the
numerical answer with the verbal
description (a) -50 (i) as likely to happen as
not (b) 0 (ii) very likely to happen, though
not certain (c) 10 (iii) wont happen (d)
50 (iv) may happen, but its not likely (e)
90 (v) will happen for sure (f) 100 (vi) made
a computational error (e) 200
2. Toss a coin 1,000 times. About how many heads
are expected?
3. Roll a die 6,000 times. About how many aces
are expected?
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Chapter 13 Exercise Sets 4. One hundred tickets
are drawn at random with replacement from one of
the two boxes shown. On each draw you will be
paid the amount shown on the ticket in dollars.
Which box is better and why? (i) 1 2 (ii) 1
3
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Chapter 13 Probability Conditional probabilities
4 suits clubs, diamonds, spades, hearts 13 cards
in each 2 through 10, jack, queen, king,
ace total of 52 cards in a deck
Example deck of cards
Shuffle the deck of cards (places them in random
order), place the top two cards face down on the
table If the second card is the queen of hearts -
win 1,000 what are your chances of winning the
?
The bet is about the second card, that is all we
need to know P(2nd card queen of hearts)
1/52 as there are 52 possible positions for the
queen, all equally likely
This is the probability of winning, written as
P(winning) P(2nd card queen of hearts)
1/52
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Chapter 13 Probability Conditional probabilities
Example deck of cards
4 suits clubs, diamonds, spades, hearts 13 cards
in each 2 through 10, jack, queen, king,
ace total of 52 cards in a deck
Shuffle the deck of cards (places them in random
order), place the top two cards face down on the
table If the second card is the queen of hearts -
win 1,000 the first card is the seven of clubs,
now what is your chance of winning?
There are 51 cards left - there is no replacement
here the 51 are in random order all possible
positions have an equal chance so the
probability is 1/51
This is a conditional probability as the first
card HAS to be the seven of clubs P(2nd card
queen of hearts 1st card is the seven of clubs)
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Chapter 13 Exercise Sets
5. Two tickets are drawn at random without
replacement from the box 1 2 3 4
(a) what is the chance that the second ticket
is 4? (b) what is the chance that the second
ticket is 4, given that the first is 2? Repeat
the question, drawing with replacement
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Chapter 13 Exercise Sets 6. A penny is tossed 5
times. (a) find the chance that the 5th toss is
a head (b) find the chance that the 5th toss is
a head, given the first 4 are tails
7. Five cards are dealt off the top of a well
shuffled deck. (a) find the chance that the 5th
card is the queen of spades (b) find the chance
that the 5th toss is the queen of spades, given
that the first 4 cards are hearts
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Multiplication Rule
600 people, each with
Draw 1 at random without replacement What are he
chances of drawing the striped can and then the
green can?
Box with 3 cans red, green, blue
200
200
200
100
100
100
100
100
100
100 of 600 people 1/2 of 1/3 1/6 16 2/3
1/3 times 1/2 1/6 the chance that 2 things will
both happen their joint probability P(1st
happening) times P(2nd happening 1st has
happened)
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Another multiplication example a well shuffled
deck of cards
4 suits clubs, diamonds, spades, hearts 13 cards
in each 2 through 10, jack, queen, king,
ace total of 52 cards in a deck
What is the probability of the first card being
the 7 of clubs and the second being the queen of
hearts?? P(1st card 7 clubs) and P(2nd card
queen of hearts) ??
P(1st card 7 clubs) 1/52 P(2nd card queen
of hearts) 1/51 Thus P(both) 1/52 times
1/51 1/2,652 or 4 in 10,000 or .04 of 1
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Another multiplication example a well shuffled
deck of cards
4 suits clubs, diamonds, spades, hearts 13 cards
in each 2 through 10, jack, queen, king,
ace total of 52 cards in a deck
What is the probability of the top two cards
being kings?? P(1st card king) and P(2nd card
king) ??
P(1st card king) 4/52 P(2nd card king)
3/51 Thus P(both) 4/52 times 3/51
12/2,652 or 1 in 200 or .5 of 1
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Chapter 13 Probability Multiplication
rule Probability of 2 things happening
P(first happening) times P(2nd happening the
1st has happened
(unconditional P of 1st times conditional P of
2nd)
Exercise set C
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Chapter 13 Exercise Sets 6. A deck is shuffled
and two cards are dealt. (a) find the chance
that the 2nd card is a heart, given that the
first card is a heart (b) find the chance
that the 1st card is a heart, and the 2nd card is
a heart
8. A die is rolled 3 times. (a) find the chance
that the first roll is an ace (b) find the
chance that the first roll is an ace, the second
roll is a deuce, and the third roll is a trey
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Chapter 13 Exercise Sets 9. A die will be
rolled 6 times. You have a choice (a) win 1.00
if at least one ace shows (b) win 1.00 if an
ace shows on all the rolls Which offers the
better chance of winning? Are they the same?
Explain
10. A coin is tossed three times. (a) what is
the chance of getting three heads? (b) what is
the chance of not getting three heads? (c) what
is the chance of getting at least one
tail? (b) what is the chance of getting at least
one head?
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Chapter 13 Independence
Two events are INDEPENDENT if P(2nd
first) is the same regardless of the outcome of
the first
Two events are DEPENDENT if P(2nd
first) depends on the outcome of the first
Example tossing a coin twice and betting that
the second will be a head P(2nd tosshead 1st
toss head) .50 P(2nd tosshead 1st toss
tail) .50 The two coin tosses are independent
of each other - the outcome on the first does
not influence the outcome of the second
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Chapter 13 Probability Multiplication
rule Probability of 2 things happening
P(first happening) times P(2nd happening the
1st has happened
(unconditional P of 1st times conditional P of
2nd)
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Draw 2 cans at random from the box with
replacement
P(1st draw red) 2/5 P(1st draw blue)
2/5 P(1st draw green) 1/5
Suppose we draw a red can on the first draw,
P(2nd blue) ?? Question are the two
events dependent or independent?? sampling
WITH REPLACEMENT, so the 2 events are independent
P(2nd draw red) 2/5 P(2nd draw
blue) 2/5 P(2nd draw green) 1/5
So P(1st draw red can) 2/5 40
and P(2nd draw blue can) 40 regardless of
the first draws outcome
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Draw 2 cans at random from the box with
replacement
P(1st draw red) 2/5 P(1st draw blue)
2/5 P(1st draw green) 1/5
Suppose we draw a red can on the first draw,
P(2nd red) ?? Question are the two
events dependent or independent?? sampling
WITH REPLACEMENT, so the 2 events are independent
P(2nd draw red) 2/5 P(2nd draw
blue) 2/5 P(2nd draw green) 1/5
So P(1st draw red can) 2/5 40
and P(2nd draw red can) 40 regardless of
the first draws outcome
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Draw 2 cans at random from the box without
replacement
P(1st draw red) 2/5 P(1st draw blue)
2/5 P(1st draw green) 1/5
Suppose we draw a red can on the first draw,
P(2nd red) ?? Question are the two
events dependent or independent?? sampling
WITHOUT REPLACEMENT, so the 2 are dependent
After the first draw we are left with
P(2nd red 1st red) 1/4 25
If the first draw is NOT red, we have
and the P(2nd red 1st is not red) 2/4 50
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Chapter 13 Independence
Drawing at random with replacement draws are
INDEPENDENT the P(of any item being selected)
remains the same
Drawing at random without replacement draws are
DEPENDENT the P(of later items being selected)
changes
Multiplication and Independence
P(1st red) 1/3 P(1st blue) 1/3 P(1st
green) 1/3
With replacement P(2nd) P(1st)
P(drawing red then blue) P(blue 2nd red 1st)
1/3 x 1/3 1/9
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Chapter 13 Independence
Drawing at random with replacement draws are
INDEPENDENT the P(of any item being selected)
remains the same
Drawing at random without replacement draws are
DEPENDENT the P(of later items being selected)
changes
Multiplication and Independence
P(1st red) 1/3 P(1st blue) 1/3 P(1st
green) 1/3
With replacement P(2nd) P(1st)
P(drawing red then blue) P(blue 2nd red 1st)
1/3 x 1/3 1/9
Without independence (without replacement) P(blue
2nd red 1st) is conditioned upon the first
drawing
1st drawing red
P(blue 2nd) 1/2
1st drawing green
P(blue 2nd) 1/2
1st drawing blue
P(blue 2nd) 0
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Chapter 13 Probability Independence and the
Multiplication rule Probability of 2 INDENDENT
events happening P(first happening) times
P(2nd happening the 1st has happened)
(unconditional P of 1st times unconditional P of
2nd)
Read section 5
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1 2 2 1 2 2
Chapter 13 Exercise Sets 11. Are color and number
independent or dependent for each of the
following boxes?
1 2 3 1 2 2
1 2 1 2 1 2
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Chapter 13 Exercise Sets 12. In each of the boxes
below, each ticket has two numbers. A ticket is
drawn at random from each - are the two numbers
dependent or independent
1 2 1 3 4 2 4 3

1 2 1 3 1 3 4 2
4 3 4 3
1 2 1 3 1 3 4 2
4 2 4 3
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Chapter 13 Exercise Sets 13. Every week you buy
a ticket at a lottery that offers a chance of one
in a million of winning. What is the chance
that you never win, even if you keep this up for
ten years?
14. A die is rolled 6 times. You win 1.00 each
time an ace shows. Find the chance that you
only win on the first roll.
15. You are presented with 3 ordinary playing
cards, face down. You choose one and also
select one at random from a separate full deck of
cards. If the two cards are from the same
suite, you win 10.00. What are your chances of
winning?