PROBABILITY
The very name calculus of probabilities is
a paradox. Probability opposed to certainty is
what we do not know, and how can we
calculate what we do not know?
H. Poincaré
Science and Hypothesis
Cosimo Classics, 2007, Chapter XI
Probability
If the Sample Space S of an experiment
consists of finitely many outcomes (points) that
are equally likely, then the probability P(A) of an
event A is
P(A) = Number of Outcomes (points) in A
Number of Outcomes (points) in S
Permutation
• A permutation is an arrangement of all or part
of a set of objects.
• Number of permutations of n objects is n!
• Number of permutations of n distinct objects
taken r at a time is
nPr = n!
(n – r)!
• Number of permutations of n objects arranged
is a circle is (n-1)!
Problem
• An encyclopedia has eight volumes. In how
many ways can the eight volumes be replaced
on the shelf?
A
64
B
16,000
C
40,000
D
40,320
Problem
• How many permutations of 3 different digits
are there, chosen from the ten digits, 0 to 9
inclusive?
A
84
B
120
C
504
D
720
Problem
• How many permutations of 4 different letters
are there, chosen from the twenty six letters
of alphabets (Repetition not allowed)?
A
14,950
B
23,751
C
358,800
D
456,976
Permutations
• The number of distinct permutations of n
things of which n1 are of one kind, n2 of a
second kind, …, nk of kth kind is
n!
n1! n2! n3! … nk!
Permutations
• The College football team consists of 1 player
from juniors, 3 players from 2nd Term, 5
players from 3rd Term and 7 players from
seniors. How many different ways can they be
arranged in a row, if only their term level will
be distinguished?
Combinations
• The number of combinations of n distinct
objects taken r at a time is
nCr
=
n!
r! (n – r)!
Problem
• In how many ways can a Committee of 5 can
be chosen from 10 people?
A
252
B
2,002
C
30,240
D
100,000
Problem
• Jamil is the Chairman of the Committee. In
how many ways can a Committee of 5 can be
chosen from 10 people, given that Jamil must
be one of them?
A
252
B
126
C
495
D
3,024
Problem
• How many different letter arrangements can
be made from the letters in the word of
STATISTICS?
Independent Probability
• If two events, A and B are independent then
the Joint Probability is
P(A and B) = P (A Π B) = P(A) P(B)
• For example, if two coins are flipped the
chance of both being heads is
1/2 x 1/2 = 1/4
Mutually Exclusive
• If either event A or event B or both events
occur on a single performance of an experiment
this
is
called
the
union
of
the
events A and B denoted as P (A U B).
• If two events are Mutually Exclusive then
the probability of either occurring is
P(A or B) = P (A U B) = P(A) + P(B)
• For example, the chance of rolling a 1 or 2 on a
six-sided die is 1/6 + 1/6 = 2/3
Not Mutually Exclusive
• If the events are not mutually exclusive then
P(A or B) = P (A U B) = P(A) + P(B) - P (A Π B)
• For example, when drawing a single card at
random from a regular deck of cards, the
chance of getting a heart or a face card (J,Q,K)
(or one that is both) is
13/52 + 12/52 – 3/52 = 22/52
Conditional Probability
• Conditional Probability is the probability of
some event A, given the occurrence of some
other event B.
• Conditional probability is written as P(A І B),
and is read "the probability of A, given B". It is
defined by
P(A І B) = P (A Π B)
P(B)
Conditional Probability
• Consider the experiment of rolling a
dice. Let A be the event of getting an
odd number, B is the event getting at
least 5. Find the Conditional Probability
P(A І B).
Conditional Probability
• Conditional Probability is the probability of
some event A, given the occurrence of some
other event B.
• Conditional probability is written as P(A І B),
and is read "the probability of A, given B". It is
defined by
P(A І B) = P (A Π B)
P(B)
Population of a Town
Male
Female
Employed
460
140
600
Unemployed
40
260
300
A: One Chosen is Employed
B: A man is Chosen
Find P(B І A)
Total
500
400
900
Members Rotary Club
Employed Unemployed Members
Male
Female
460
140
600
40
260
300
E - 36
U - 12
48
A: One Chosen is Employed
B: Member of Rotary Club
Find P(B І A)
Find P(B І A’)
Total
500
400
900
Independent Events
Two
events,
A
and
B,
are independent if the fact that
A
occurs
does
not
affect
probability of B occurring.
P(A and B) = P(A) · P(B)
the
Independent Events
A coin is tossed and a single 6-sided
die is rolled. Find the probability of
landing on the head side of the coin
and rolling a 3 on the die.
Dependent Events
Two events are dependent if the
outcome or occurrence of the first affects
the outcome or occurrence of the second
so that the probability is changed.
Dependent Events - Example
A card is chosen at random from a
standard deck of 52 playing cards. Without
replacing it, a second card is chosen. What
is the probability that the first card chosen
is a queen and the second card chosen is a
jack?
Theorem of Total Probability
P(B) = P(A1 Π B) + P(A2 Π B) + P(A3 Π B) + …
+ P(Ak Π B)
Bayes’ Rule
If the events B1, B2, B3, … . Bk constitute a
partition of the Sample Space S such that
P(Bi) = 0, for i = 1, 2, … , k, then for any event
A in S such that P( A ) = 0,
P (Br | A) = P (Br Π A)
∑ P (Bi Π A)
= P(Br ) P (A l Br)
∑ P(Bi ) P (A l Bi)
Bayes’ Rule - Example
In a certain Assembly Plant, three
machines B1, B2, and B3, make 30%, 45%,
and 25%, respectively of the product. It is
known from the past experience that 2%,
3% and 2% of the products made by each
machine respectively are defective. Now, we
suppose that a finished product is randomly
selected. What is the probability that it is
defective?
Bayes’ Rule - Example
In a certain Assembly Plant, three
machines B1, B2, and B3, make 30%, 45%,
and 25%, respectively of the product. It is
known from the past experience that 2%,
3% and 2% of the products made by each
machine respectively are defective. Now, we
suppose that a finished product is randomly
selected. What is the probability that it is
defective?
Bayes’ Rule - Example
If the Product was chosen randomly
and found to be defective. What is the
Probability that it was made by
machine B3?
Complementation Rule
For an event A and its complement A’ in
a Sample Space S, is
P(A’) = 1 – P(A)
Example - Complementation Rule
5 coins are tossed. What is the
probability that:
a. At least one head turns up
b. No head turns up
Problem 1
Three screws are drawn at random from
a lot of 100 screws, 10 of which are
defective. Find the probability that the
screws drawn will be non-defective in
drawing:
a. With Replacement
b. Without Replacement
Problem 3
If we inspect paper by drawing 5
sheets without replacement from every
batch of 500. What is the probability of
getting 5 clean sheets although 2% of
the sheets contain spots?
Problem 5
If you need a right-handed screw from
a box containing 20 right-handed screws
and 5 left-handed screw. What is the
probability that you get at least one right
handed screws in drawing 2 screws with
replacement?
Problem 7
What gives the greater possibility of
hitting some targets at least once:
a. Hitting in a shot with probability ½
and firing one shot
b. Hitting in a shot with probability 1/4
and firing two shots
Problem 11
In rolling two fair dice, what is the
probability of obtaining equal number or
numbers with an even product?
Problem 13
A motor drives an electric generator.
During a 30 days period, the motor
needs repair with 8% and the generator
needs repair with probability 4%. What
is the probability that during a given
period, the entire apparatus (consisting
of a motor and a generator) will need
repair?
Problem 15
• If a certain kind of tire has a life
exceeding 25,000 miles with probability
0.95. What is the probability that a set of
4 of these tires on a car will last longer
than 25,000 miles?
• What is the probability that at least one
of these tires on a car will lost longer than
25,000 miles?
Problem 17
A pressure control apparatus contains 4
values. The apparatus will not work unless
all values are operative. If the probability
of failure of each value during some
interval of time is 0.03, what is the
corresponding probability of failure of the
apparatus?
QUIZ # 2
32 (Cptr) A – 9 OCT 2012
• If you need a right-handed screw from a
box containing 20 right-handed screws
and 5 left-handed screw. What is the
probability that you get at least one right
handed screws in drawing 2 screws
without replacement ? (Rows 1 & 3)
• In rolling a fair dice, what is the probability
of obtaining a sum greater than 4 but not
exceeding 7 ? (Rows 2 & 4)
QUIZ # 2
32 (Cptr) B – 8 OCT 2012
A pressure control apparatus contains 4
valves. The apparatus will not work unless
all valves are operative. If the probability of
failure of each valve during some interval
of time is 0.03, what is the corresponding
probability of failure of the apparatus?