Download RANDOM EXPERIMENT OUTCOME or ELEMENT SAMPLE SPACE

RANDOM EXPERIMENT
STATS 103.3
- a process of observation such that
- there is more than one
possible observation, and
- the observation that occurs
cannot be predicted with
certainty.
PROBABILITY
Definitions
Models for Probability
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OUTCOME or ELEMENT
Random Experiment
• one of the possible observations
when a random experiment is
performed.
• May denote such an element as e
e.g. A card is selected at random
from a standard deck of cards.
The card selected could be any
one of the 52 cards in the deck.
It is impossible to predict with
certainty which card will be
obtained.
• e.g. Each of the 52 cards in the deck is
an element or outcome.
e1 = A♠ (Ace of Spades is observed)
e2 = K♦ (King of Diamonds is observed)
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STANDARD DECK OF 52 CARDS
SAMPLE SPACE
• the set of all possible outcomes of
a random experiment. (denote it S )
•
S = { e1, e2, e3, … eN }
• e.g. S = {the 52 cards in a deck}
• When a coin is tossed,
S = { H, T }
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A♣
A♦
A♥
A♠
2♣
2♦
2♥
2♠
3♣
3♦
3♥
3♠
4♣
4♦
4♥
4♠
5♣
5♦
5♥
5♠
6♣
6♦
6♥
6♠
7♣
7♦
7♥
7♠
8♣
8♦
8♥
8♠
9♣
9♦
9♥
9♠
10 ♣
10 ♦
10 ♥
10 ♠
J♣
J♦
J♥
J♠
Q♣
Q♦
Q♥
Q♠
K♣
K♦
K♥
K♠
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EVENT
TRIAL
• one repetition of a simple
experiment.
• a subset of the sample space.
–Denote events as E, A etc.
• Examples:
– Selecting one card. A second trial is
then the selection of another card.
– Rolling a die. A second trial is a
second roll.
– Toss a coin. Toss it again.
• e.g. H = {card selected is a Heart}
M = {number on die is a multiple of 3}
= { 3, 6 }
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SIMPLE EVENT
• an event involving one single
outcome or element of the
sample space.
• Thus, E = { e }
e.g. A = {card is the Seven of Hearts}
= { 7♥}
B = { Heads }
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CERTAIN or SURE Event
- This event MUST happen when
the experiment is performed.
- It is equivalent to the whole
sample space S
- e.g. E = {card is red or black} = S
L = {number on die is less than
10} = S
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IMPOSSIBLE or NULL Event
What is PROBABILITY?
- This event CANNOT happen when
the experiment is performed.
PROBABILITY is the likelihood or
chance of the occurrence of an
event or outcome.
- It is denoted φ
- It contains NO elements of the sample
space S
e.g. One card is selected at random from
a standard deck of 52 cards. What is
the probability that an Ace is chosen?
- e.g. G = {card is green} = φ
N = {number on die is negative} = φ
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BASIC PROPERTIES
NOTATION
If E is an event in a sample
space S, then the probability of
the event E is denoted
P[ E ] or Pr[ E ]
1. 0 ≤ P [ E ] ≤ 1
e.g. P[ F ] could denote the probability
of obtaining a Face card when one card
is drawn at random from a deck.
3. For any two distinct elements e and f of S,
2.
for every event E.
(probability is a number
between zero and one)
P [S ] = 1
(The probability of the sure event is one.
The total probability available is one.)
P [ e or f ] = P [ e ] + P [ f ]
(for distinct outcomes, probabilities add)
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Models for Probability
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Examples
1. EQUALLY LIKELY
or Classical Probability
1. What is the probability that a card drawn at
random from a standard deck is a Face card?
#( F ) 12 3
P [F] =
= = = 0.2308
# ( S) 52 13
If the sample space S of a random
experiment contains N = # (S) equally
likely outcomes, and if an event E
contains n = # (E) of these outcomes,
then
# (E ) n
P [E ] =
=
# (S ) N
2. What is the probability that a multiple of 3 is
observed when a die is rolled?
#( M) 2 1
P[ M] =
= = = 0.3333
#( S) 6 3
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Models for Probability
Models for Probability
2. RELATIVE FREQUENCY Probability
2. RELATIVE FREQUENCY Probability
If a random experiment is repeated for
M trials, and if this results in m
occurrences of an event E, the relative
frequency of occurrence of E is
f =
The probability of the event E is the
limiting value of the relative frequency
as the number of trials M increases
without bound (goes to infinity). That is
m
M
P [E ] =
{}
lim m
M→∞ M
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Examples
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Models for Probability
1. What is the probability that a thumbtack will
fall with its point “up” when dropped on the
floor?
2. What is the probability that a skilled dart
player will be able to hit the region of the
dart board at which he/she aims?
3. EMPIRICAL (or Statistical) Probability
The probability of the event E is
approximated by the relative frequency
of its occurrence based on a number of
trials of a random experiment.
P [E ] ≅
m
M
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Examples
Models for Probability
1. Students in a certain class were classified
according to gender and eye colour.
Female
Male
Blue
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22
Brown
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33
Hazel
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4. SUBJECTIVE Probability
Other
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After analyzing and studying all
available information relevant to the
problem, personal judgment is used.
2. Lengths in cm of fish caught in a lake last
summer were summarized as follows.
Length
Number
< 20
72
20 – 40
124
40 – 60
156
22
> 60
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e.g. What is the probability of snow
tomorrow?
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Models for Probability
5. ODDS and Probability
Odds indicate the number of equally
likely chances for an event
happening versus the number of
equally likely chances against it
happning.
If the odds in favour of an event E
occurring are m:n, then the
probability of event E happening is
m
m+n
e.g. The probability of a student
successfully completing STATS 103.3
is
P [E ] =
P [ E] =
e.g. The odds are 7:2 that a student will
successfully complete STATS 103.3 .
m
7
7
=
= = 0.7778
m+ n 7 +2 9
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If the odds against an event E
occurring are m:n, then the
probability of event E happening is
n
P [E ] =
m+n
e.g. In a race, the odds against a certain
horse are quoted as 5:3 . The indicated
probability of this horse winning the
race is
n
3
3
P [E] =
=
= = 0.3750
m+ n 5 + 3 8
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Probability to Odds
If the probability of an event E occurring
is P[ E ], then the odds in favour of E
are determined by starting with the
ratio
C
P [ E ] : P E 
and reducing it to the form m:n in which
m and n are the smallest possible
positive integers having no common
factors.
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Example: The probability of a certain
Huskie team winning its next home
game is 0.84 . What are the odds in
favour of this happening?
Probability to Odds
If the probability of an event E occurring
is P[ E ], then the odds in favour of E
are determined by starting with the
ratio
C
P [ E ] : P  E C  ⇒ 0.84 : 0.16
P [ E ] : P E 
⇒ 84 : 16 ⇒ 21 : 4
The odds against are just the reverse
ratio
The odds against them winning are
then
4 : 21
P  E C  : P [ E ]
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