Probability

1
Probability
Chapter 3
2
3.1

• Basic Concepts of Probability

3
Basic Concepts of Probability

• If you go to a supermarket and select 5 lbs of
  apples at 0.79 per lb, you can easily predict
  the amount you will be charged (5 0.79 3.95)
  at the checkout counter. The amount charged for
  such purchases is a Deterministic Phenomenon. It
  can be predicted exactly on the basis of the
  information given.
• On the other hand, consider the problem faced by
  the produce manager of the supermarket, who must
  order enough apples to have on hand each day
  without knowing exactly how many pounds customer
  will buy during the day. The customers demand is
  an example of Random Phenomenon. The study of
  probability is concerned with such random
  phenomenon. Even though we cant be certain,
  whether or not a given result will occur, we can
  often obtain a good measure of its likelihood or
  probability.
• The theory of mathematical probability originated
  from game of chance.

4
Probability Experiments
A probability experiment is an action through
which specific results (counts, measurements or
responses) are obtained.
Example Rolling a die and observing the number
that is rolled is a probability experiment.
The result of a single trial in a probability
experiment is the outcome.
The set of all possible outcomes for an
experiment is the sample space.
Example The sample space when rolling a die has
six outcomes. 1, 2, 3, 4, 5, 6
5
Events
An event consists of one or more outcomes and is
a subset of the sample space.
Example A die is rolled. Event A is rolling an
even number.
A simple event is an event that consists of a
single outcome.
Example A die is rolled. Event A is rolling an
even number. This is not a simple event because
the outcomes of event A are 2, 4, 6.
6
Classical Probability
Classical (or theoretical) probability is used
when each outcome in a sample space is equally
likely to occur. The classical probability for
event E is given by
Example A die is rolled. Find the probability
of Event A rolling a 5.
There is one outcome in Event A 5
7
Empirical Probability
Empirical (or statistical) probability is based
on observations obtained from probability
experiments. The empirical frequency of an event
E is the relative frequency of event E.
Example A travel agent determines that in every
50 reservations she makes, 12 will be for a
cruise. What is the probability that the next
reservation she makes will be for a cruise?
8
Law of Large Numbers
As an experiment is repeated over and over, the
empirical probability of an event approaches the
theoretical (actual) probability of the event.
9
Example The following frequency distribution
represents the ages of 30 students in a
statistics class. What is the probability that a
student is between 26 and 33 years old?
P (age 26 to 33)
10
Subjective Probability
Subjective probability results from intuition,
educated guesses, and estimates.
Example A business analyst predicts that the
probability of a certain union going on strike is
0.15.
Range of Probabilities Rule The probability of an
event E is between 0 and 1, inclusive. That is
0 ? P(A) ? 1.
11
Complementary Events
The complement of Event E is the set of all
outcomes in the sample space that are not
included in event E. (Denoted E' and read E
prime.)
P(E) P (E' ) 1
P(E) 1 P (E' )
P (E' ) 1 P(E)
Example There are 5 red chips, 4 blue chips,
and 6 white chips in a basket. Find the
probability of randomly selecting a chip that is
not blue.
12
ODDS

• If an event is twice as likely to occur than not
  to occur, we say that the odds are 2 to 1.
• In betting, the word Odds is also used to
  denote the ratio of the wager of one party to
  that of another. If a gambler says that he/she
  will give 3 top 1 odds on the occurrence of an
  event, the gambler means that he/she is willing
  to bet 3 against 1 (or perhaps 30 against 10)
  that the event will occur. That is, if you win,
  you make 3 for each 1 bet.

13
Odds
14

• Example Let event E is rolling a pair of dice
  and getting a double.
• Green
• 1 2 3 4
  5 6
• 1 (1,1) (1,2) (1,3) (1,4)
  (1,5) (1,6)
• 2 (2,1) (2,2) (2,3) (2,4)
  (2,5) (2,6)
• R 3 (3,1) (3,2) (3,3)
  (3,4) (3,5) (3,6)
• e 4 (4,1) (4,2) (4,3)
  (4,4) (4,5) (4,6)
• d 5 (5,1) (5,2) (5,3)
  (5,4) (5,5) (5,6)
• 6 (6,1) (6,2) (6,3) (6,4)
  (6,5) (6,6)
• The above table is called Product Table.

15
Note Probability and odds are two different
quantities. Odds represent a ratio of two
numbers not a fraction. Where as, probability is
a fraction and could be also represented as a
decimal number.
16
Review Examples for Probability and Odds

• Example 1 A card is drawn at random from a deck
  of 52 cards. Find the following probabilities and
  odds of the events

17
(No Transcript)
18

• c. P(a face card and a club) 3/52
• ?
• K, Q, J of club
• d. P(a face card or a club) 22/52
• ?
• 12 face cards 10 other clubs
• Note in each case 0 P(A) 1 is true.

19
3.2

• Conditional Probability and the Multiplication
  Rule

20
Conditional Probability
A conditional probability is the probability of
an event occurring, given that another event has
already occurred.
P (B A)
Probability of B, given A
Example There are 5 red chip, 4 blue chips, and
6 white chips in a basket. Two chips are
randomly selected. Find the probability that the
second chip is red given that the first chip is
blue. (Assume that the first chip is not
replaced.)
Because the first chip is selected and not
replaced, there are only 14 chips remaining.
21
Conditional Probability
Example 100 college students were surveyed and
asked how many hours a week they spent studying.
The results are in the table below. Find the
probability that a student spends more than 10
hours studying given that the student is a male.
The sample space consists of the 49 male
students. Of these 49, 16 spend more than 10
hours a week studying.
22
Independent Events
Two events are independent if the occurrence of
one of the events does not affect the probability
of the other event. Two events A and B are
independent if P (B A) P (B) or if P (A B)
P (A). Events that are not independent are
dependent.
Example Decide if the events are independent or
dependent.
Selecting a diamond from a standard deck of
cards (A), putting it back in the deck, and then
selecting a spade from the deck (B).
The occurrence of A does not affect the
probability of B, so the events are independent.
23
Multiplication Rule
The probability that two events, A and B will
occur in sequence is P (A and B) P (A) P (B
A). If event A and B are independent, then the
rule can be simplified to P (A and B) P (A) P
(B).
Example Two cards are selected, without
replacement, from a deck. Find the probability
of selecting a diamond, and then selecting a
spade.
Because the card is not replaced, the events are
dependent.
P (diamond and spade) P (diamond) P (spade
diamond).
24
Multiplication Rule
Example A die is rolled and two coins are
tossed. Find the probability of rolling a 5,
and flipping two tails.
P (5 and T and T ) P (5) P (T ) P (T )
25
3.3

• The Addition Rule

26
Mutually Exclusive Events
Two events, A and B, are mutually exclusive if
they cannot occur at the same time.
A and B are not mutually exclusive.
A and B are mutually exclusive.
27
Mutually Exclusive Events
Example Decide if the two events are mutually
exclusive.
Event A Roll a number less than 3 on a die.
Event B Roll a 4 on a die.
These events cannot happen at the same time, so
the events are mutually exclusive.
28
Mutually Exclusive Events
Example Decide if the two events are mutually
exclusive.
Event A Select a Jack from a deck of cards.
Event B Select a heart from a deck of cards.
Because the card can be a Jack and a heart at the
same time, the events are not mutually exclusive.
29
The Addition Rule
The probability that event A or B will occur is
given by P (A or B) P (A) P (B) P (A and
B ). If events A and B are mutually exclusive,
then the rule can be simplified to P (A or B) P
(A) P (B).
Example You roll a die. Find the probability
that you roll a number less than 3 or a 4.
The events are mutually exclusive.
P (roll a number less than 3 or roll a 4)
P (number is less than 3) P (4)
30
The Addition Rule
Example A card is randomly selected from a deck
of cards. Find the probability that the card is
a Jack or the card is a heart.
The events are not mutually exclusive because the
Jack of hearts can occur in both events.
P (select a Jack or select a heart)
P (Jack) P (heart) P (Jack of hearts)
31
The Addition Rule
Example 100 college students were surveyed and
asked how many hours a week they spent studying.
The results are in the table below. Find the
probability that a student spends between 5 and
10 hours or more than 10 hours studying.
The events are mutually exclusive.
P (5 to10) P (10)
P (5 to10 hours or more than 10 hours)
32
Review Examples

• Addition Rule
• P(A or B) P(A) P(B) P(A and B)
• ? ?
• either A or B or both A and B A and B
  both occur at the same time
• If event A and B are disjoint or mutually
  exclusive (can not both occur at the same time)
  then P(A and B) 0, and P(A or B) P(A) P(B)

33
Review Examples

• General Multiplication Rule
• P(A and B) P(A) P(B/A)
• Or P(B and A) P(B) P(A/B)
• Where P(B/A) represent the conditional
  probability of event B given that event A has
  already occurred.
• Independent Events Two events A and B are
  independent if knowing that one occurs does not
  change the probability that the other occurs. If
  event A and B are independent then, P(B/A) P(B)
  and
• P(A and B) P(A) P(B)

34
Examples

• Example 1 Out of 36 people applying for the
  job, 20 are men and 16 are women. Eight of the
  men and 12 of the women have Ph.D.s. If one
  person is selected at random for the first
  interview, find the probability that the one
  chosen has a Ph. D.
• First organize the information
• M W Total
• Ph.D 8 12 20
• Not Ph.D. 12 4 16
• Total 20 16 36

35
Example Continued

• a. P(PhD) 20/36
• b. the one chosen is a woman and has a Ph.D.
• P(W and PhD) 12/36
• c. the one chosen is a woman or has a Ph.D.
• P(W or PhD) P(W) P(PhD) P(W and PhD)
• 16/36 20/36 12/36
• 24/36 2/3

36
Examples

• Example 2. Of the 20 television programs to be
  aired this evening, Marc plans to watch one,
  which he will pick at random by throwing a dart
  at TV schedule. If 8 of the programs are
  educational, 9 are interesting, and 5 are both
  educational and interesting, find the probability
  that the show he watches will have at least one
  of these attributes.
• If E represent educational and I represent
  interesting, then
• P(E) 8/20, P(I) 9/20, and P(E and I)
  5/20

37
Examples

• 3. The probability that a person selected at
  random did not
• graduate from high school is 0.25. If three
  people are
• selected at random, find the probability
  that
• a. all three do not have a high school
  diploma.
• Since each person is independent ,
• P(all three do not have a high school diploma)
  (.25)3
• b. all three have a high school diploma.
• P(all three have a high school diploma) (1 -
  .25)3
• (.75)3
• Note Events in Part a and b above are not
  complement of each other.

38
Example Continued

• 3c. at least one has high school diploma.
• P(at least one has HS diploma) 1 P(none
  have HS diploma)
• 1 (.25)3 .9844
• In general
• P(event happening at least once) 1 P(event
  does not happen)
• Ex. If a family has six children, find the
  probability that at
• least one boy in the family?
• There are 26 64 equally likely outcomes.
• Since the complement of at least one boy is
  all girls
• P(at least one boy) 1 P(all girls)
• 1 1/26 1 1/64 63/64

39
Examples

• 4. The World Wide Insurance Company found that
• 53 of the residents of a city had
  homeowners insurance with the company. Of
  these clients, 27 also had car insurance with
  the company. If a resident is selected at random,
  find the probability that the resident has both
  homeowners and car insurance with the World Wide
  Insurance Company.
• Given P(homeowners insurance) 53
• P(car insurance / homeowners insurance) 27
• P(homeowners and car insurance) (.53)
  (.27)
• .1431

40
Conditional probability
Note rule for independence If a sample size is
no more than 5 of the size of the population,
treat the selection as being independent (even if
the selections are made without replacement).
41
3.4

• Counting Principles

42
Counting Methods

• 1. By Listing Simply by identifying all the
  different items in a list and counting them.
• Example 1 Flipping three coins S HHH,
  HHT, ., TTT, 8 possibilities.
• Example 2 Rolling a pair of dice, there are 36
  possibilities.
• Example 3 Consider a team of 4 peopleT A, B,
  C, D all equally likely
• a. How many ways they can select a Captain
  for the team?
• Since all team members are equally likely, there
  are 4 ways.

43
Counting Methods

• b. Number of ways they can select a Captain and
  a Leader (no one can hold both positions)
• Use a product table Leader
• A B C D
• A AB AC AD
• B BA BC BD
• C CA CB CD
• D DA DB DC
• Total 12 possibilities.

C a p t a i n
44
Example continued

• Here AB represent that A is a captain and B is a
  leader, where as BA represent that B is a captain
  and A is a leader order is important
  (Permutation)
• Can also use Slot filling method to find out
  total
• Possibilities in this case.
• 4 ways to select a Captain and then only 3
  ways to select the Leader
• 4 3 12 ways (and translate to
  multiplication)
• c. Select a sub team of 2 members for a game
  (order is not important ie. AB BA) can be done
  in 6 ways AB, AC, AD, BC, BD, and CD.

45
Fundamental Counting Principle
If one event can occur in m ways and a second
event can occur in n ways, the number of ways
the two events can occur in sequence is m n.
This rule can be extended for any number of
events occurring in a sequence.
Example A meal consists of a main dish, a side
dish, and a dessert. How many different meals
can be selected if there are 4 main dishes, 2
side dishes and 5 desserts available?
of main dishes
of side dishes
of desserts
4
5
2
?
?

40
There are 40 meals available.
46
Fundamental Counting Principle
Example Two coins are flipped. How many
different outcomes are there? List the sample
space.
Start
1st Coin Tossed
2 ways to flip the coin
Heads
Tails
2nd Coin Tossed
Heads
2 ways to flip the coin
Heads
Tails
Tails
There are 2 ? 2 4 different outcomes HH, HT,
TH, TT.
47
Fundamental Counting Principle
Example The access code to a house's security
system consists of 5 digits. Each digit can be 0
through 9. How many different codes are
available if a.) each digit can be repeated? b.)
each digit can only be used once and not repeated?
a.) Because each digit can be repeated, there
are 10 choices for each of the 5 digits.
10 10 10 10 10 100,000 codes
b.) Because each digit cannot be repeated, there
are 10 choices for the first digit, 9 choices
left for the second digit, 8 for the third, 7 for
the fourth and 6 for the fifth.
10 9 8 7 6 30,240 codes
48
Permutations
A permutation is an ordered arrangement of
objects. The number of different permutations of
n distinct objects is n!.
n! n (n 1) (n 2) (n 3) 3 2 1
Example How many different surveys are required
to cover all possible question arrangements if
there are 7 questions in a survey?
7! 7 6 5 4 3 2 1 5040 surveys
49
Permutation of n Objects Taken r at a Time
The number of permutations of n elements taken r
at a time is
in the group
taken from the group
Example You are required to read 5 books from a
list of 8. In how many different orders can
you do so?
50
Distinguishable Permutations
The number of distinguishable permutations of n
objects, where n1 are one type, n2 are
another type, and so on is
Example Jessie wants to plant 10 plants along
the sidewalk in her front yard. She has 3 rose
bushes, 4 daffodils, and 3 lilies. In how many
distinguishable ways can the plants be
arranged?
51
Combination of n Objects Taken r at a Time
A combination is a selection of r objects from a
group of n things when order does not
matter. The number of
combinations of r objects selected from a group
of n objects is
in the collection
taken from the collection
Example You are required to read 5 books from a
list of 8. In how many different ways can you do
so if the order doesnt matter?
52
Application of Counting Principles
Example In a state lottery, you must correctly
select 6 numbers (in any order) out of 44 to win
the grand prize.
a.) How many ways can 6 numbers be chosen from
the 44 numbers? b.) If you purchase one lottery
ticket, what is the probability of winning the
top prize?
a.) b.) There is only one winning ticket,
therefore,