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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?