T3 probability
- 1. QM Probability Page 1 QM/T3
- 2. Concept of Probability You are rolling a die. What are the possible outcomes (value of the top face) if the die is rolled only once? Outcomes are either 1 or 2 or 3 or 4 or 5 or 6. Chance of occurring ‘1’ is one out of six. Page 2 QM/T3
- 3. Concept of Probability Probability that outcome of roll of a die is ‘1’ is 1/6. P(Die roll result = 1) = 1/6 Simple EVENT Outcome Probability 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 Page 3 QM/T3
- 4. Concept of Probability What is the probability that outcome of roll of a die is an even number. Thisoutcome can occur if the roll result is 2 or 4 or 6. i.e. 3 ways. Number of all possible result is 6. P(Even number) = 3/6 = 0.5 Similarly probability of odd number is 3/6. Page 4 QM/T3
- 5. Concept of Probability A 1 3 5 2 4 6 B P(A) = 3/6 = 0.5 P(B) = 3/6 = 0.5 P(A) + P(B) = 1 NOTE •A & B are mutually exclusive. P(A) = 1 – P(B) •A & B are mutually exhaustive. •Sum probability of all outcome is 1. •Probability is always ≥ 0. Page 5 QM/T3
- 6. What is Sample Spaces Collection of all possible outcomes. All six faces of a die: All 52 cards in a deck: Page 6 QM/T3
- 7. Events Simple event Outcome from a sample space with one characteristic. e.g.: A red card from a deck of 52 cards Page 7 QM/T3
- 8. Visualizing Events Red cards 26 Black cards 26 Total cards 52 P(A red card is drawn from a deck) = 26/52 = 0.5 Page 8 QM/T3
- 9. Events Compound event Involves at least two outcomes simultaneously. e.g.:An ace that is also red from a deck of cards. Page 9 QM/T3
- 10. Visualizing Events Ace Others Total Red 2 24 26 Black 2 24 26 Total 4 48 52 P(An Ace and Red) = 2/52 Page 10 QM/T3
- 11. Impossible Events Impossible event e.g.: One card drawn is ‘Q’ of Club & diamond Also known as ‘Null Event’ Page 11 QM/T3
- 12. Joint Probability P(An ‘Ace’ and ‘Red’ from a deck of cards) Ace Others Total Red 2 24 26 Black 2 24 26 Total 4 48 52 A = ‘Ace’ P(A & B) = ? B = ‘Red’ Page 12 QM/T3
- 13. Joint Probability The probability of a joint event, A and B: P(A and B) = P(A ∩ B) B A A&B No. of outcomes from A and B = Total No. of possible outcomes Page 13 QM/T3
- 14. Compound Probability Probability of a compound event, A or B: P(A or B) = P(A U B) B A A&B No. of outcomes from A only or B only or Both = Total No. of possible outcomes Page 14 QM/T3
- 15. Compound Probability P(A or B) = P(A U B) = P(A) + P(B) – P(A ∩ B) All Ace All Red 4 2 26 4 26 2 7 = + - = Addition Rule 52 52 52 13 Page 15 QM/T3
- 16. Compliment Set In roll of a die Set of all outcomes = {1,2,3,4,5,6}. If A = {1} then Ac has {2,3,4,5,6}. If B = Set of even outcomes = {2, 4, 6} then Bc has {1,3, 5}. In a class if A = Set of students that have passed, then Ac is = Set of students that have not passed. Page 16 QM/T3
- 17. Computing Joint Probability A = Card drawn from deck is ‘Ace’ Ac = Card drawn from deck is not ‘Ace’ B = Card drawn from deck is ‘Red’ B c = Card drawn from deck is not ‘Red’ Event Total Event B Bc A 2 2 4 Ac 24 24 48 Total 26 26 52 Page 17 QM/T3
- 18. Computing Joint Probability A = Card drawn from deck is ‘Ace’ Ac = Card drawn from deck is not ‘Ace’ B = Card drawn from deck is ‘Red’ B c = Card drawn from deck is not ‘Red’ Event c Total Event B B A A∩B A∩Bc A Ac Ac ∩ B Ac ∩ B c Ac Total B Bc Page 18 QM/T3
- 19. Computing Joint Probability Event Total Event B Bc A P(A ∩ B) P(A ∩ B c) P(A) Ac P(Ac ∩ B) P(Ac ∩ B c) P (Ac) Total P(B) P(B c) 1 Page 19 QM/T3
- 20. Conditional Probability Finding probability of an event A, given that event B has occurred. This means that we need to find out probability of occurrence of ‘Ace’ given that the card drawn is ‘Red’. Event Total Event B Bc 2 A 2 2 4 = c 26 A 24 24 48 Total 26 26 52 Page 20 QM/T3
- 21. Conditional Probability Theprobability of event A given that event B has occurred P(A ∩ B) = This is known as P(B) ‘Conditional Probability’ and denoted as: P(A l B) Page 21 QM/T3
- 22. Multiplication Rule P(A ∩ B) P(A ∩ B) = x P(B) P(B) = P(A l B) x P(B) = P(B l A) x P(A) Page 22 QM/T3
- 23. Statistical Independence Events A and B are independent if the probability of one event, A, is not affected by another event, B P(A l B) = P(A) P(B l A) = P(B) P(A and B) = P(A) x P(B) Page 23 QM/T3
- 24. Bayes’s Theorem P(B l A) x P(A) P(A l B) = P(B) Page 24 QM/T3
- 25. Bayes’s Theorem P(B l A) x P(A) P(A l B) = P(B) P(B l A) x P(A) = P(B ∩ A) + P(B ∩ Ac) P(B l A) x P(A) = P(B l A) x P(A) + P(B l Ac) x P(Ac) Page 25 QM/T3
- 26. Bayes’s Theorem (General) P(A l Bi) x P(Bi) P(Bi l A) = P(A l B1).P(B1) +…+ P(A l Bk).P(Bk) ) Page 26 QM/T3
- 27. Example Fifty percent of borrowers repaid their loans. Out of those who repaid, 40% had a college degree. Ten percent of those who defaulted had a college degree. What is the probability that a randomly selected borrower who has a college degree will repay the loan? Repaid loan Not repaid loan Total College degree 0.2 0.05 0.25 No college degree 0.3 0.45 0.75 Total 0.5 0.5 1.0 Page 27 QM/T3
- 28. Example Fifty percent of borrowers repaid their loans. Out of those who repaid, 40% had a college degree. Ten percent of those who defaulted had a college degree. What is the probability that a randomly selected borrower who has a college degree will repay the loan? CD - College degree NCD – No College degree P(RL | CD) = ? RL – Repaid Loan NRL – Not Repaid Loan Page 28 QM/T3
- 29. Example Fifty percent of borrowers repaid their loans. Out of those who repaid, 40% had a college degree. Ten percent of those who defaulted had a college degree. What is the probability that a randomly selected borrower who has a college degree will repay the loan? CD - College degree P(RL | CD) = ? NCD – No College degree RL – Repaid Loan 0.4 0.5 NRL – Not Repaid Loan P(CD | RL) P(RL) P(CD | RL) P(RL) + P(CD | NRL) P(NRL) 0.4 0.5 0.1 0.5 Page 29 QM/T3
- 30. Class Exercise 1. P(A) = 0.25, P(B) = 0.4, P(A|B) = 0.15 Find out P(AUB). 2. Probability of two independent events A and B are 0.3 and 0.6 respectively. What is P(A∩B)? 3. P(A∩B) = 0.2 ; P(A∩C) = 0.3 ; P(B|A) + P(C|A) = 1; What is P(A)? Page 30 QM/T3
- 31. Class Exercise 4. A jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. a) What is the probability of choosing a red marble? 5. You are tossing a coin three times. a) What is the probability of getting two tails? b) What is the probability of getting at least 2 heads? Page 31 QM/T3
- 32. Class Exercise 6. A plant has 3 assembly lines that produces memory chips. Line1 produces 50% of chips (defective 4%), Line2 produces 30% of chips (defective 5%), Line3 produces the rest (defective 1%). A chip is chosen at random from produced lot. a) What is the probability that it is defective? b) Given that the chip is defective, what is the probability that it is from Line2? Page 32 QM/T3
- 33. Class Exercise 7. An urn initially contains 6 red and 4 green balls. A ball is chosen at random and then replaced along with two additional ball of same colour. This process is repeated. a) What is the probability that the 1st and 2nd ball drawn are red and 3rd is green? b) What is the probability of 2nd ball drawn is red? Page 33 QM/T3
- 34. Class Exercise 8. Two squares are chosen at random on a chessboard. What is the probability that they have a side in common? 9. An anti aircraft gun can fire four shots at a time. If the probabilities of the first, second, third and the last shot hitting the enemy aircraft are 0.7, 0.6, 0.5 and 0.4, what is the probability that four shots aimed at an enemy aircraft will bring the aircraft down? Page 34 QM/T3