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Elementary Statistics and
Inference
22S:025 or 7P:025
Lecture 16
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Elementary Statistics and
Inference
22S:025 or 7P:025
Chapter 13
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13. What Are the Chances?
A.
Introduction – Probabilities of events can be defined by
observed frequencies of events, by a mathematical model,
and by a personal conviction (bayesian). Our book presents
probability concepts based on a frequency theory.
The chance that an event will happen is based on the
percentage of time it is expected to happen, when the basic
process is repeated over and over again, independently and
under the same conditions.
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1
13. What Are the Chances? (cont.)
The chances that an event can happen is between 0%
and 100%.
0.0 ≤ P ( X ) ≤ 1.00
Example: When you roll a die, the chances (probability)
that a “six” will show is 1/6 = 16.67% of the time, or
P (6) =
number of ways six can occur
total number of ways die can fall
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13. What Are the Chances? (cont.)
Example: Box contains 3 blue marbles, and 2 red
marbles. If a marble is selected, what are the
chances it would be “red”?
P( red ) =
3
number of red marbles
= = 60%
total number of marbles in box 5
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13. What Are the Chances? (cont.)
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2
13. What Are the Chances? (cont.)
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13. What Are the Chances? (cont.)
Drawing With Replacement
Suppose you have a “box” with three tickets labeled 1, 2,
or 3.
1
2
3
Shake the box, and select a ticket – each ticket has an
equal chance of being selected.
P( X ) =
1
3
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13. What Are the Chances? (cont.)
Drawing Without Replacement
Shake the box with the three tickets – select a ticket,1 the
chance of any of the three tickets being selected is 3 .
S the
Set
h ticket
i k aside,
id then
h select
l
a second
d ticket.
i k
Th
The
chance of selecting a particular value is :
P( X ) =
number of ways to obtain a particular ticket
total number of remaining tickets
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3
13. What Are the Chances? (cont.)
Example: For three tickets labeled 1, 2, 3
With replacement, the chance of selecting a three on
the second draw is 1/3.
‰
‰
Without replacement, the chance of selecting a three
on the second draw, given that it was not selected on
the first draw is ½.
Exercises : Exercise Set A (pp. 225-226)
1, 2, 3, 4, 5
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13. What Are the Chances? (cont.)
2)
A coin is tossed 1000 times. About how many heads
would you expect?
P( H ) =
1
⎛1⎞
⇒ 1000⎜ ⎟ = 500
2
⎝ 2⎠
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13. What Are the Chances? (cont.)
3)
A die is rolled 6,000 times. About how many aces
(number 1) would you expect?
P (1) =
4)
1
⎛1⎞
⇒ 6000⎜ ⎟ = 1000
6
⎝6⎠
In five-card draw poker, the chance of being dealt a full
house (one pair and three of a kind) is 0.14 of 1% (i.e.,
.0014). If 10,000 hands are dealt, about how many will
be a full house?
P ( Full ) = .0014 ⇒ 10,000(.0014) = 14
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13. What Are the Chances? (cont.)
5)
One hundred tickets will be drawn at random with
replacement from one of the two boxes shown below.
On each draw you will be paid the amount shown on
the ticket in dollars. Which box is better and why?
ii
i
1
2
1
3
100[P (1)(1) + P(2)(2)]
100[ P (1)(1) + P(3)(3)]
⎡⎛ 1 ⎞
⎛1⎞ ⎤
100⎢⎜ ⎟(1) + ⎜ ⎟(2)⎥
⎝ 2⎠ ⎦
⎣⎝ 2 ⎠
100[1.5] = $150
⎡⎛ 1 ⎞
⎛1⎞ ⎤
100⎢⎜ ⎟(1) + ⎜ ⎟(3)⎥
⎝ 2⎠ ⎦
⎣⎝ 2 ⎠
100[2] = $200
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13. What Are the Chances? (cont.)
B.
Conditional Probabilities
A conditional probability describes the chances of an
event happening, given that a different event has
already occurred.
occurred
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13. What Are the Chances? (cont.)
Example: For a deck of cards (4 suites, with 13 cards
in each suite) – shuffle the deck, and select 2 cards.
You win $1.00 if the second card is the queen of
hearts!
a)
What is chance of winning the dollar?
P (Q ) =
1
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(unconditi onal chance)
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13. What Are the Chances? (cont.)
b)
You turn first card over. It is seven of clubs.
What is chance of winning the dollar?
P(Queen of H 1st card is 7 of Clubs) = P(Q 7) =
ƒ
1
(conditional chance)
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This illustrates conditional probability.
Exercise Set B (page 227) 1, 3, 4
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13. What Are the Chances? (cont.)
C.
Multiplication Rule
„
The chances that two events will happen equals the
chance that the first event will occur, multiplied by the
chance that the second event will occur.
Example: A box has 3 different marbles, all of same
size and shape. One Red, one Blue, and one White.
Find probability that the first draw will be red
followed by the second draw will be
1 1 1
white - 3 × 2 = 6
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13. What Are the Chances? (cont.)
Example: A deck of 52 cards. Find probability that the
first draw will be Ace of Spades, and second draw
will be a “7”.
P ( A) • P (7) =
1 4
4
• =
= .0015
52 51 2,652
Example:
E
l A coin
i iis ttossed
d ttwo titimes. Wh
Whatt iis chance
h
(probability) that the first outcome is a “head”, and
the second outcome is a “tail”?
P ( H ) • P (T ) =
1 1 1
• =
2 2 4
Exercise Set C (pp. 229-230) #2, 3, 4, 5, 6
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6
13. What Are the Chances? (cont.)
D.
Independence
•
Two events are independent if the chances
(probability) of the second given the first outcome are
the same
same, no matter the outcome of the first event
event.
Otherwise the two events are dependent.
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13. What Are the Chances? (cont.)
Example: A coin is tossed two times. If the coin lands
as a head on the second toss you win a dollar.
‰
‰
If the first toss is heads, what is your chance of
winning the dollar?
If the first toss is tails, what is your chance of
winning the dollar?
Outcomes are independent P ( H ) =
1
2
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13. What Are the Chances? (cont.)
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13. What Are the Chances? (cont.)
Example: Two draws will be made at random from a
box with replacement.
1
1
2
2
3
a) Suppose first draw is “1”, what is chance of “2” on
second draw?
2
P ( 2) = = .40
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13. What Are the Chances? (cont.)
b)
Suppose draws are made without replacement, find
chance first draw is a “1”, followed by second draw
equal to a “2”.
2
= .40, but for second draw we assume a "1" was selected on
5
2 1
first draw, so P(2) = = = .50
4 2
P(1) =
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13. What Are the Chances? (cont.)
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13. What Are the Chances? (cont.)
See examples on pages 230-232.
Exercise Set D (page 232) #2, 3, 4, 6, 7
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13. What Are the Chances? (cont.)
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13. What Are the Chances? (cont.)
Review Exercises (pp. 235-236) #4, 6, 7, 8, 9, 10
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13. What Are the Chances? (cont.)
10.
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