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Probabilit y and Statistics

Probabilit y and Statistics. Probability. The likelihood of the occurrence of an event resulting from statistical experiments . Probability of an event. It is the sum of the weights of all sample points in A. Hence. Probabilit y and Statistics. Probability.

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Probabilit y and Statistics

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  1. Probability and Statistics Probability • The likelihood of the occurrence of an event resulting from statistical experiments Probability of an event • It is the sum of the weights of all sample points in A. Hence

  2. Probability and Statistics Probability Example 3.1: A coin is tossed twice. What is the probability that at least one head occurs? • If the coin is balanced, each of the outcomes in the sample space is equally likely to occur. Therefore, the weight of each outcome is:

  3. Probability and Statistics Probability Example 3.1: A coin is tossed twice. What is the probability that at least one head occurs? • The event that contains at least one head is:

  4. Probability and Statistics Probability Example 3.2: A die is loaded in such a way that an even number is twice as likely to occur as an odd number. If E is the event that a number less than 4 occurs on a single toss of the die, find P(E). • chance of an odd number • chance of an even number

  5. Probability and Statistics Probability Example 3.2: A die is loaded in such a way that an even number is twice as likely to occur as an odd number. If E is the event that a number less than 4 occurs on a single toss of the die, find P(E).

  6. Probability and Statistics Probability Example 3.2: A die is loaded in such a way that an even number is twice as likely to occur as an odd number. If E is the event that a number less than 4 occurs on a single toss of the die, find P(E).

  7. Probability and Statistics Probability Example 3.3: from example 3.2 let A be the event that an even number turns up and let B be the event that a number divisible by 3 occurs. Find P(AB) and P(AB)

  8. Probability and Statistics Probability Example 3.3: from example 3.2 let A be the event that an even number turns up and let B be the event that a number divisible by 3 occurs. Find P(AB) and P(AB)

  9. Probability and Statistics Theorem 2.9 • If an experiment can result in any one of N different equally likely outcomes, and if exactly n of these outcomes correspond to event A, then the probability of event A is

  10. Probability and Statistics Theorem 2.9 Example 3.4: A mixture of candies contains 6 mints, 4 toffees, and 3 chocolates. If a person makes a random selection of one of these candies, find the probability of getting a mint a toffee or a chocolate

  11. Probability and Statistics Theorem 2.9 Example 3.5: In a poker hand consisting of 5 cards, find the probability of holding 2 aces and 3 jacks

  12. Probability and Statistics Theorem 2.9 Example 3.5: In a poker hand consisting of 5 cards, find the probability of holding 2 aces and 3 jacks

  13. Probability and Statistics Theorem 2.10 • If A and B are any two events, then Corollary 1 • If A and B are mutually exclusive, then Corollary 2 • If A1, A2, A3, … An are mutually exclusive, then

  14. Probability and Statistics Theorem 2.10 • If A and B are any two events, then Corollary 3 • If A1, A2, A3, … An is a partition of a sample space S, then

  15. Probability and Statistics Theorem 2.11 • For three events A, B, and C,

  16. Probability and Statistics Theorem 2.11 Example 3.6: The probability that Paula passes mathematics is 2/3, and the probability that she passes English is 4/9. If the probability of passing both courses is 1/4, what is the probability that Paula will pass at least one of these courses?

  17. Probability and Statistics Theorem 2.11 Example 3.7: What is the probability of getting a total of 7 or 11 when a pair of dice are tossed? • Let A – event that a total of 7 occurs • Let B – event that a total of 11 occurs

  18. Probability and Statistics Theorem 2.11 Example 3.7: What is the probability of getting a total of 7 or 11 when a pair of dice are tossed? • Events A and B are mutually exclusive since a total of 7 or 11 cannot both occur on the same toss

  19. Probability and Statistics Theorem 2.11 Example 3.8: If the probabilities are, respectively, 0.09, 0.15, 0,21, and 0.23 that a person purchasing a new automobile will choose the color green, white, red or blue, what is the probability that a given buyer will purchase a new automobile that comes in one of those colors?

  20. Probability and Statistics Theorem 2.12 • If A and A’ are complementary events, then Example 3.8: If the probabilities that an automobile mechanic will service 3, 4, 5, 6, 7, 8 or more cars on any given work day are, 0.12, 0.19, 0.28, 0.24, 0.10, and 0.07, what is the probability that he will service at least 5 cars on his next day at work?

  21. Probability and Statistics Theorem 2.12 • If A and A’ are complementary events, then Example 3.8: If the probabilities that an automobile mechanic will service 3, 4, 5, 6, 7, 8 or more cars on any given work day are, 0.12, 0.19, 0.28, 0.24, 0.10, and 0.07, what is the probability that he will service at least 5 cars on his next day at work?

  22. Probability and Statistics Example 3.9: A box contains 500 envelops of which 75 contain $100 in cash, 150 contain $25, and 275 contain $10. An envelop may be purchased for $25. Find the probability that the first envelop purchased contains less than $100.

  23. Probability and Statistics Example 3.9: A box contains 500 envelops of which 75 contain $100 in cash, 150 contain $25, and 275 contain $10. An envelop may be purchased for $25. Find the probability that the first envelop purchased contains less than $100.

  24. Probability and Statistics Example 3.10: The probability that an American industry will locate in Munich is 0.7, the probability that it will locate in Brussels is 0.4, and the probability that it will locate in either Munich or Brussels or both is 0.8. What is the probability that the industry will locate in Both cities? Neither city? Given:

  25. Probability and Statistics Example 3.10: The probability that an American industry will locate in Munich is 0.7, the probability that it will locate in Brussels is 0.4, and the probability that it will locate in either Munich or Brussels or both is 0.8. What is the probability that the industry will locate in Both cities? Neither city? Given:

  26. Probability and Statistics Example 3.11: An automobile manufacturer is concerned about a possible recall of their best-selling four-door sedan. If there were a recall, there is 0.25 probability that a defect is in a brake system, 0.18 in the transmission, 0.17 in the fuel system, and 0.40 in some other area. What is the probability that the defect is the brakes or the fueling system if the probability of defects in both systems simultaneously is 0.2? What is the probability that there are no defects in either the brakes or the fueling system?

  27. Probability and Statistics Example 3.11: An automobile manufacturer is concerned about a possible recall of their best-selling four-door sedan. If there were a recall, there is 0.25 probability that a defect is in a brake system, 0.18 in the transmission, 0.17 in the fuel system, and 0.40 in some other area. What is the probability that the defect is the brakes or the fueling system if the probability of defects in both systems simultaneously is 0.15?

  28. Probability and Statistics Example 3.11: An automobile manufacturer is concerned about a possible recall of their best-selling four-door sedan. If there were a recall, there is 0.25 probability that a defect is in a brake system, 0.18 in the transmission, 0.17 in the fuel system, and 0.40 in some other area. What is the probability that the defect is the brakes or the fueling system if the probability of defects in both systems simultaneously is 0.15?

  29. Probability and Statistics Example 3.11: An automobile manufacturer is concerned about a possible recall of their best-selling four-door sedan. If there were a recall, there is 0.25 probability that a defect is in a brake system, 0.18 in the transmission, 0.17 in the fuel system, and 0.40 in some other area. What is the probability that there are no defects in either the brakes or the fueling system?

  30. Probability and Statistics Conditional Probability • The probability of an event B occurring when it is known that some event A has occurred Definition • The conditional probability of B, given A, denoted by P(B|A), is defined by if P(A) > 0

  31. Probability and Statistics Example 3.12: Suppose that our sample space S is the population of adults in a small town who have completed the requirements for a college degree. We shall categorize them according to sex and employment status: One of these individuals is to be selected at random for a tour throughout the country to publicize the advantages of establishing new industries in the town.

  32. Probability and Statistics Example 3.12: Suppose that our sample space S is the population of adults in a small town who have completed the requirements for a college degree. We shall categorize them according to sex and employment status: P (M) = probability that a man is chosen P (E) = probability that the one chosen is employed

  33. Probability and Statistics Example 3.12: Suppose that our sample space S is the population of adults in a small town who have completed the requirements for a college degree. We shall categorize them according to sex and employment status:

  34. Probability and Statistics Example 3.13: The probability that a regularly scheduled flight departs on time is P(D) = 0.83; the probability that it arrives on time is P(A) = 0.82; and the probability that it departs and arrives on time is P(D  A) = 0.78. Find the probability that a plane Arrives on time given that it departed on time Departs on time given that it arrived on time Arrives on time given that it departed late

  35. Probability and Statistics Example 3.13: The probability that a regularly scheduled flight departs on time is P(D) = 0.83; the probability that it arrives on time is P(A) = 0.82; and the probability that it departs and arrives on time is P(D  A) = 0.78. Find the probability that a plane Arrives on time given that it departed on time Departs on time given that it arrived on time Arrived on time given that it departed late

  36. Probability and Statistics Example 3.13: The probability that a regularly scheduled flight departs on time is P(D) = 0.83; the probability that it arrives on time is P(A) = 0.82; and the probability that it departs and arrives on time is P(D  A) = 0.78. Find the probability that a plane Arrives on time given that it departed on time Departs on time given that it arrived on time Arrived on time given that it departed late

  37. Probability and Statistics Example 3.13: The probability that a regularly scheduled flight departs on time is P(D) = 0.83; the probability that it arrives on time is P(A) = 0.82; and the probability that it departs and arrives on time is P(D  A) = 0.78. Find the probability that a plane Arrives on time given that it departed on time Departs on time given that it arrived on time Arrived on time given that it departed late

  38. Probability and Statistics Example 3.14: from example 3.2 - A die is loaded in such a way that an even number is twice as likely to occur as an odd number. Let B be the event of getting a perfect square, and A be the event of getting a number greater than 3. Find the probability of getting a perfect square given that the number is greater than 3.

  39. Probability and Statistics Example 3.14: from example 3.2 - A die is loaded in such a way that an even number is twice as likely to occur as an odd number. Let B be the event of getting a perfect square, and A be the event of getting a number greater than 3. Find the probability of getting a perfect square given that the number is greater than 3.

  40. Probability and Statistics Independent Event • The probability of an event B occurring when it is known that some event A has occurred • Example 3.15: 2 cards are drawn in succession from an ordinary deck with replacement. • Let A – the first card is an ace • Let B – the second card is a spade

  41. Probability and Statistics Independent Event • The probability of an event B occurring when it is known that some event A has occurred • Example 3.15: 2 cards are drawn in succession from an ordinary deck with replacement. • Let A – the first card is an ace • Let B – the second card is a spade

  42. Probability and Statistics Multiplicative Rules Theorem 2.13 • If in an experiment the events A and B can both occur, then

  43. Probability and Statistics Theorem 2.13 • Example 3.16: A fuse box contains 20 fuses, of which 5 are defective. If 2 fuses are selected at random and removed from the box in succession without replacing the first, what is the probability that both fuses are defective? • Let A – the first fuse is defective • Let B – the second fuse is defective

  44. Probability and Statistics Theorem 2.13 Example 3.17: One bag contains 4 white balls and 3 black balls, and a second bag contains 3 white balls and 5 black balls. One ball is drawn from the first bag and placed unseen in the second bag. What is the probability that a ball now drawn from the second bag is black?

  45. Probability and Statistics Theorem 2.13 Example 3.17: One bag contains 4 white balls and 3 black balls, and a second bag contains 3 white balls and 5 black balls. One ball is drawn from the first bag and placed unseen in the second bag. What is the probability that a ball now drawn from the second bag is black?

  46. Probability and Statistics Theorem 2.14 • Two events A and B are independent if and only if Example 3.18: A small town has one fire engine and one ambulance available for emergencies. The probability that a fire engine is available when needed is 0.98, and the probability that the ambulance is available when called is 0.92. In an event of an injury resulting from a burning building, find the probability that both the ambulance and the fire engine will be available.

  47. Probability and Statistics Theorem 2.15 • If in an experiment, the events A1, A2, A3,…Ak can occur, then • If the events A1, A2, A3,…Ak are independent, then

  48. Probability and Statistics Theorem 2.15 Example 3.19: Three cards are drawn in succession, without replacement, from an ordinary deck of playing cards. Find the probability that the event A1 A2  A3 occurs, where A1 is the event that the first card is a red ace, A2 is the event that the second card is a 10 or jack, and A3 is the event that the third card is greater than 3 but less than 7.

  49. Probability and Statistics Theorem 2.15 Example 3.19: Three cards are drawn in succession, without replacement, from an ordinary deck of playing cards. Find the probability that the event A1 A2  A3 occurs, where A1 is the event that the first card is a red ace, A2 is the event that the second card is a 10 or jack, and A3 is the event that the third card is greater than 3 but less than 7.

  50. Probability and Statistics Theorem 2.16: Total Probability • If the events B1, B2, …, Bk constitute a partition of the sample S such that P(Bi)  0 for i = 1,2,…k, then for any event A of S,

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