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CHAPTER 2
Introduction to probability Written by:
Samah sindi
Shafiaa alhodaira
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Random Experiment:
It is an experiment whose possible outcomes are known but cannot be predicted with certainty.
Ex(1). Tossing a coin once.
Ex(2). Rolling a die once.
Ex(3). Drawing a card from a deck.
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Sample Space:
It is a set whose elements represent all possible outcomes of a random
experiment. It is usually denoted by S.
Ex(4). Tossing a coin twice.
S={HH, TH, HT, TT}
Ex(5). Rolling a die once.
S={1 , 2 , 3 , 4 , 5, 6}
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Event :
It is a subset of a sample space. It is usually denoted by A , B , C , …..
Ex(6). Rolling a die once.
A is the Event of getting an odd number.
A={ 1 , 3 , 5}
B is the Event of getting a number greater than 3.
B={ 4 , 5 , 6}
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Ex(7): Tossing a coin 3 times.
i- Find the event of getting one head.
A= { HTT , THT , TTH }
ii- Find the event of getting at least one head.
B= { HTT, TTH, THT, HHT, HTH, THH, HHH } iii- Find the event of getting at most 2
heads.
C= { TTT, TTH, THT, HTT, HHT, HTH, THH }
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Operations on Events
Complement of event : Consist of all outcomes in the
sample space S that are not in A, denoted by A
Ac , it means A does not occur Intersection : The event that consisting of all outcomes that are both in event A and B, denoted by A B , AB.
Union : The event that consisting of all outcomes that are either in A or in B or in both A and B, denoted by AB
Operations on Events
Sets:
𝑨 − 𝑩 = 𝑨 ∩ 𝑩 (
A
occurs andB
does not occur) 𝑩 − 𝑨 = 𝑨 ∩ 𝑩 (
B
occurs andA
does not occur) 𝑨 ∩ 𝑩 (neither
A
norB
occurs) 𝑨 ∪ 𝑩 (either not
A
or notB
occurs) Demorgan’s Laws:• 𝑨 ∪ 𝑩 = 𝑨 ∩ 𝑩 .
• 𝑨 ∩ 𝑩 = 𝑨 ∪ 𝑩 .
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Kinds of Events
Simple event:
is an event that contains only one element
(outcome). For example, if a die is rolled once and a 6 turned up then the event A={6} is
called simple event.
Compound event:
is an event that contains more than one
element. For example, if a die is rolled once and the event C is an event of getting odd numbers, so C={1,3,5} is called a compound event.
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Kinds of Events
Sure event:
is an event that contains all outcomes of the sample space. For example, the event of
getting a number when a die is rolled once.
The event E={1,2,3,4,5,6} is called a sure event.
Null or impossible event:
is an event that does not contain any element (outcome) of the sample space, and is denoted by ∅. For example, the event of getting a letter when a die is rolled once. Or the event of
getting a number when a coin is flipped.
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Two events A and B are called disjoint or mutually exclusive events if A and B cannot occur at the same time , A ∩ B=∅
Ex(8): Rolling a die one time getting an odd number and an even number.
Ex(9): Tossing a coin one time getting head and tail.
Mutually Exclusive :
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(1)Subjective Approach:
It depends on experience and the amount of available information.
Ex(10): A doctor based on his opinion said that the probability of the success for the new medicine is 80%.
Definitions of probability
Definitions of probability
(2) Classical Approach:
It depends on assuming all outcomes are equally likely. In this case: P(A)=n(A)/n(S) n(A), number of elements in the event A.
n(S), number of elements in the sample space S.
Equally likely:
Two events are said to be equally likely if they have the same probability.
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Definitions of probability
Ex(11): A die is rolled one time, find the following probabilities:
i- Getting an even number.
S={1, 2, 3, 4, 5, 6} A={2 ,4 ,6}
P(A)= 3/6 = ½
ii- Getting number divisible by 3.
B={ 3, 6}
P(B)= 2/6 = 1/3
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Definitions of probability
(3) Empirical Approach:
It depends on repeating the experiment
“n” times and outing the number “m” of occurrence of the event A, and taking
P(A)=m/n, for sufficiently large n.
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Definitions of probability
Ex(12): Human blood is grouped into 4 types in the following table:
If we choose a person find the following probabilities:
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type frequency
A 7
B 6
O 13
AB 4
Total 30
Definitions of probability
i- He has type O blood.
P(O)= 13/30
ii- He has type A or B blood.
P(A or B)= P(A)+P(B)
= 7/30+6/30= 13/30 iii- He hasn’t type AB blood.
P( not AB)= P(A or B or O) = P(A)+P(B)+P(O) = 7/30+6/30+13/30=26/30
Or P( not AB)=1-P(AB)=1-4/30=26/30
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Basic Theorems
1- 0 ≤ P(A) ≤ 1 2- P(S) = 1
3- P(A)+P(𝑨𝒄)= 1 → P(A)= 1- P(𝑨𝒄) 4- For any events A , B in S:
P(A-B)=P(A) – P(A∩B)
5- For any events A , B in S, such that A ⊂ B:
P(A) ≤ P(B)
6- (i) If A, B are not mutually exclusive events in S:
P( A or B)= P( A∪B)= P(A)+P(B)-P(A∩B)
(ii) If A,B are mutually exclusive events in S:
P( A or B)= P( A∪B)= P(A)+P(B)
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Basic Theorems
7- (i) If A , B, and C are any events in S:
P( A∪B∪C )=P(A)+P(B)+P(C)-P(A∩B)-P(A∩C) -P(B∩C)+P(A∩B∩C)
(ii) If A , B, and C are mutually exclusive events in S:
P( A∪B∪C )=P(A)+P(B)+P(C)
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Ex (13): Three horses A, B, C are in a race, A is twice as likely to win as B and B is twice likely to win as C, then P(A) equals…
P(A)=2P(B) P(B)=2P(C) P(A)=4P(C)
P(A)+P(B)+P(C)=1 4P(C)+2P(C)+P(C)=1 7P(C)=1 → P(C)=1/7
P(A)=4/7 then P(B)=2/7
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Ex(14): Let A and B defined on the sample space S such that:
P(A)=0.3 P(B)=0.25 P(A∩B)=0.07 Find P( A ∪ B)?
P( A∪B)= P(A)+P(B)-P(A∩B)
= 0.3 + 0.25 – 0.07= 0.48
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