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. PoincaréScience 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)

Employed Unemployed TotalMale 460 40 500

Female 140 260 400600 300 900

A: One Chosen is EmployedB: A man is Chosen

Find P(B І A)

Population of a Town

Employed Unemployed Members TotalMale 460 40 E - 36 500

Female 140 260 U - 12 400600 300 48 900

A: One Chosen is EmployedB: Member of Rotary Club

Find P(B І A)Find P(B І A’)

Members Rotary Club

Independent Events

Two events, A and B,

are independent if the fact that

A occurs does not affect the

probability of B occurring.

P(A and B) = P(A) · P(B)

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 upb. No head turns up

Problem 1Three 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 Replacementb. 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 13A 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 # 232 (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 # 232 (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?