Basic Definition and Axioms of Probability

1
Chapter 2

• Basic Definition andAxioms of Probability

• Sample Space and Event
• Relative Frequency of an Event
• Probability of an Event
• Basic Properties of Probability
• Probability of the Repetition of Identical and
  Independent Events

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Objective

• To understand the concept of sample space and the
  events of an experiment
• To familiarize with the concept of the
  probability of an event
• To know more about the properties of probability

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Sample Space ( S )

• Sample space is a set of all possible outcomes of
  an experiment which is denoted by S.
• Note an experiment can have different sample
  spaces depending on what we are interested to
  observe.
• Types of sample space
• Finite, countable ---- discrete part
• Infinite, uncountable ---- continuous part

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Event ( E )

• An event is a subset of the sample space of an
  experiment which is denoted by E
• Hence
• An event can be the individual outcomes, the
  entire sample space, or the null set.
• Any designated collection of possible outcomes of
  an experiment constitutes an event.
• That means if the outcome of an experiment is one
  of the constitute members of the event, an event
  has occurred.
• Elementary event ( e ) a single possible element
  of S
• Hence -------for both finite and countable S

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Event ( E )

• Some general denotations of events
• ( The union of the events E and F )
• or ( The intersection of the events E
  and F )
• Ø ( The null event )
• ( The events of not E )
• Properties of events
• Commutative laws
• Associative laws
• Distributive laws

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DeMorgans Law
7

• Ex 2.1 Considering the experiments of tossing
  two coins, we can give the following sample
  space.
• S1 HH, HT, TH, TT
• S2 (2, 0), (1, 1), (0, 2)
• S3 A, D Aalike, Ddifferent
• Ex 2.2 If an experiments consists of measuring
  sleeping hours of a special bird for its whole
  life, what is the sample space of it?
• Ans
• The sample space consists of all nonnegative
  real numbers.

number of tail
number of head
8

• Ex 2.3 A box initially contains five different
  tickets numbered from 1 to 5 two tickets are
  drawn from the box and designated as the winners.
  What are the sample spaces if the prizes given
  to the winners are different and are the same
  respectively.
• Ans
• Ep to select 2 tickets from 5 tickets
• For the prizes are the same
• S (1, 2), (1, 3), (1, 4), (1, 5), (2, 3),
  (2, 4), (2, 5), (3, 4), (3, 5), (4, 5)
• For the prizes are different
• S (1, 2), (1, 3), (1, 4), (1, 5), (2, 3),
  (2, 4), (2, 5), (3, 4), (3, 5), (4, 5), (2, 1),
  (3, 1), (4, 1), (5, 1), (3, 2), (4, 2), (5, 2),
  (4, 3), (5, 3), (5, 4)
• where for (x, y), x----first winner, y----second
  winner
• Ex 2.4 There is an experiment of throwing a
  black die and a white die. Give the sample space
  to a event of this experiment.
• Ans
• Ep to throw black and white dies
• E sum is 7

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Relative Frequency of an Event

• The relative frequency of an event represents the
  frequency of the number of times in n repetitions
  of an experiment in the event E.
• Hence, relative frequency of an event
• Properties of relative frequency
• ---if E and F are disjoint (EFØ)

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Probability of an Event

• The probability of an event E denoted by P(E) is
  defined as the limiting proportion of times that
  E occurs.
• In another words, it is the limiting frequency
  of E.
• Axiom 1
• Axiom 2
• Axiom 3 for , and
• In general or
• ( If E1, E2, , are mutually exclusive events )

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Probability of an Event

• Equally likely outcomes all the elementary
  events of a discrete sample space have the same
  probability.
• Let E , S be size of E and size of S
  respectively
• Note Use the generalized basic principle of
  counting to solve these problems

12

• Ex 2.5 Find the probability of getting odd
  numbers when throwing a fair die
• Ans
• Method 1
• Method 2

13

• Ex 2.6 A painted cube is cut into 1000 small
  cubes of the same size and an arbitrary cube is
  select. Find the probability of that the cube has
  only three painted faces.
• Ans

• Ex 2.7 From a box containing n good items and m
  defections, 5 items are randomly chosen for the
  quality test. If first k items are good, find the
  probability that the next item is good.
• Ans
• Selecting the (k1)th item from the remaining

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Basic properties of probability

• P1
• Proof As By Axiom 2
• As By Axiom 3

S
EC
E
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P2Proof

S
EC
E
F
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P3Proof
S
E
F
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P4 The Inclusion and Exclusion Principle (IE
Principle)Proof Using the
Mathematical Induction prove it.(Proof of P3 as
reference)
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P5Proof
If
form a partition of S, then for any
The set F1, F2, Fn forms a partition of S so
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Probability of the repetition of identical and
independent events

• What is the probability if we want exactly r
  times of an elementary events to occur within n
  trials?
• We can let p be the probability of the
  individual elementary event to occur and let q be
  the probability of the individual elementary
  event not to occur.
• As there are altogether nCr sequence outcomes
  of having n trials, so the probability of having
  exactly r times of an elementary events to occur
  is
• E.g. Find the probability of getting 2 sixes
  out of 4 fair die throws.
• E (6, 6, x, x), (6, x, 6, x), (6, x, x, 6),
  (x, 6, 6, x), (x, 6, x, 6), (x, x, 6, 6),
• (Note x here means a non-6 number is obtained
  )
• In general, when we need to consider the truth
  value of an event which depends on a repetition
  of identical independent elementary events, the
  Binomial Coefficient would be useful.

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Ex. 2.8 Suppose we have 3 sets of letters and
envelopes. If we put the 3 letters into the
envelopes randomly, what is the probability of at
least one letter put into its correct envelope.

• Ans
• Method 1

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Ans. Method 2 By the IE Principle
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Ex. 2.9 A fair die is thrown until a tail comes
up for the first time. What is the probability
for that happening on an even numbered throws.
Ans
23
Ex. 2.10 In a ten-question true or false
examination paper, find the probability that a
student gets a 80 percent or above correct by
random choosing. Would the chance be improved
with more questions?Ans

• So the probability that a student gets a grade
  of 80 percent or better is 0.0547
• Therefore, the chance will not be improved
  with more questions.

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Ex. 2.11 On the average, how many times must a
die be thrown until one gets a 6? p1/6 probabili
ty of a 6 on a given trial. 1-pq
5/6 probability of other than 6 on a given
trial On average
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Another method
q
p
q
p
q
p
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Ex. 2.12 There are r red socks and b black socks,
(1) What is the probability that two socks drawn
at random will have the same color? (2) If the
probability that both are red is 1/2, How small
the number of socks can be? Solution
(1) (2)
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Ex. 2.13 Mark gets off work at random times
between 3 and 5 pm. His mother lives uptown, his
girl friend downtown. Downtown trains run past
Marks stop at, say, 300, 310, 320, .etc, and
uptown trains at 301, 311, 321. He takes the
first subway that comes in either direction and
eats dinner with the one he is first delivered to
. His mother complains that he never comes to see
her, but he says she has a 50-50 chance. Whats
wrong?
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Ex. 2.14

• If A chooses one spinner, B chooses one of the
  remaining.
• One lands on the highest number is the winner.
• Would you rather be player A or B?
• Pr agtb Pr a9 Pr a5, b3 Pr a5,
  b4
• 1/31/9 1/9 5/9
• Pr bgta 1-5/9 4/9
• Pr agtc Pr a9 Pr a5, c2 1/31/9
  4/9
• Pr cgta 5/9
• Pr bgtc Pr b8 Pr b4, c2 Pr b3,
  c2
• 1/31/91/9 5/9
• Pr cgtb 4/9
• Pr BgtA 5/9
• Bs decision based on A Pr AgtB 4/9
• You want to be B.

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Ex. 2.15 If 2 balls are randomly drawn from a
bowl containing 6 white and 5 black balls, what
is the probability that one of the drawn balls is
white and the other is black? How about
two blacks? How about two whites?
Check
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Ex. 2.16 A 5-card poker hand is said to be a
full house if it consists of 3 cards of the same
denomination and 2 cards of the same
denomination. (Full house 3 of a kind plus a
pair). What is the probability that one is dealt
a full house? Total possible hands,
equally likely. There are
different combinations of say 2 tens and 3 jacks.
There are 13 different choices for the kind of
pair, after a pair has been chosen, there are 12
other choices for the remaining 3 cards. The
probability of a full house is
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• Ex. 2.17 In the game of bridge, the entire deck
  of 52 cards is dealt out to 4 players.
• What is the probability that one of the players
  receives all 13 spades?
• What is the probability that each player receives
  1 ace?
• There are possible divisions of
  cards among the 4 distinct players. If one player
  receives all 13 spades, there are
  possible divisions of other cards.
• Ans
• (b) Except aces, there are
  possible divisions of other cards
• Ans

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Ex. 2.18 A basketball team consists of 6 black
and 4 white players. If players are divided into
roommates at random, find the probability that
there will be 2 black-white roommate
pairs. Before that, How many ways to divide 4
players into roommates? a b c
d Therefore there are
ways
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How many ways to divide 10 players into
roommates There are ways to choose 2
black players ways to choose 2 white
players (B1 W1) (B2 W2) 2 ways to form
roommates (B1 W2) (B2 W1)
Remaining black
Remaining white
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Ex. 2.19 A total of 36 members of a club play
tennis, T 36 Tennis, B 18 badminton S 28
squash TnS 22 both tennis and squash TnB
12 both tennis and badminton SnB 9 both squash
and badminton TnSnB all three sports. How
many members of this club play at least one of
these sports? N(T U S U B) N(T) N(S) N(B)
N(TS) N(TB) N(SB) N(TSB)
362818-22-12-94 43 How many play Tennis
only? N(T)-N(TB)-N(TS)N(TSB) 36 - 12 -22 4
6 Basketball only N(B)-N(TB)-N(SB)N(TSB)
18 - 12 - 9 4 1 Squash only
N(S)-N(TS)-N(BS)N(TSB) 28 - 22 - 9 4 1
T
B
8
6
1
4
18
5
1
S
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• Ex. 2.20 Matching Problem
• Suppose that each of N men at a party throws his
  hat into the center of the room, and then each
  man randomly selects a hat.
• What is the probability that none of men selects
  his own hat?
• What is the probability that exactly k of men
  select their own hats?
• Solution
• Let Ei be the event that ith man selects his own
  hat, then

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• Pr none of the men selects his own hat
• Pr exactly k of the N men selects their own
  hats
• when k0, it is e-1

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• Ex. 2.21 If 10 married couples are seated at
  random at a round table, compute the probability
  that no wife sit next to her husband.
• Let Ei be the event that ith couple sit next to
  each other. We want to compute
• There are 19! ways of arranging 20 people
• n men sitting next to their wives.
• We have to arrange 20-2nn20-n entities around
  a round table. There are (20-n-1)!(19-n)!
  arrangements
• Each of the n married couples can be arranged
  next to each other in one of two possible ways,
  there are 2n(19-n)! arrangements.

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Ex. 2.22 Runs. An athletic team finished a
season with n wins and m losses. What is the
probability that there are exactly r runs of
wins? LLLL WWW LLL WW..W . W.W L.L
y1 x1 y2
x2 xr
yr1 All possible combinations
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A sequence of events En, n 1 is an increasing
sequence A sequence of events En, n 1
is an decreasing sequence If En is
increasing or decreasing sequence of events, then
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Ex. 2.23 Probability and Paradox
Suppose that we posses an infinitely large urn ad
an infinite collection of balls labeled ball
number 1, number 2, number 3 and so on.
Consider an experiment performed as follows. At
1 minute to 1200pm, balls numbered 1 through 10
are placed in the urn, and ball number 10 is
withdrawn. (Assume the withdrawal takes no time.)
At ½ minute to 1200pm, balls numbered 11
through 20 are placed in the urn, and ball number
20 is withdrawn. At ¼ minute to 1200pm, balls
numbered 21 through 30 are placed in the urn, and
ball number 30 is withdrawn. At 1/8 minute to
1200pm, and so on. The question of interest
is, how many balls are in the urn at 1200pm?
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1
2
10
11
20
30
.
......
.
1 minute
1/2 minute
1/4 min
1/8 min
1200pm
1
9
1
9

1
9


11
19
11
19


.
21
29

10
20
30
How many balls are in the urn at 1200pm?
Infinite All balls except 10n, n 1.
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However, let us change the experiment and suppose
that at 1 minute to 1200pm, balls numbered 1
through 10 are placed in the urn, and ball number
1 is withdrawn at ½ minute to 1200pm, balls
numbered 11 through 20 are placed in the urn, and
ball number 2 is withdrawn at ¼ minute to
1200pm, balls numbered 21 through 30 are placed
in the urn, and ball number 3 withdrawn at 1/8
minute to 1200pm, balls numbered 31 through 40
are placed in the urn, and ball number 4 is
withdrawn, and so on. For this new experiment
how many balls are in the urn at 1200pm?
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1 minute
1/2 minute
1/4 min
1/8 min
1200pm
4
10
3
10

2
10


11
20
11
20


.
21
30

1
2
3
How many balls are in the urn at 1200pm? 0
Ball n would have been withdrawn at (1/2)n-1
minutes before 1200pm
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Ex. 2.24 If a ball is randomly selected and
withdrawn, how many balls left in the urn at
1200pm? En is the event that ball 1 is still in
the urn after the first n withdrawn
En, n 1 is a decreasing sequence.
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Pr ball 1 is in the urn at 1200 PM
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Let Fi be the event that ball i in the urn at
1200 pm, i 11, 12, 1320
Thus with probability 1, the urn is empty at
1200pm