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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 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: 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: 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 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). 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). 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(AB) and P(AB) 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(AB) and P(AB) 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 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) a mint b) a toffee or a chocolate 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 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 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 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 Probability and Statistics Theorem 2.11 For three events A, B, and C, 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? 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 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 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? 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? 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? 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. 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. 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 a) Both cities? Given: b) Neither city? 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 a) Both cities? Given: b) Neither city? 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. a) 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? b) What is the probability that there are no defects in either the brakes or the fueling system? 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. a) 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? 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. a) 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? 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. b) What is the probability that there are no defects in either the brakes or the fueling system? 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 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: Male Female Total Employed Unemployed 460 40 140 260 600 300 Total 500 400 900 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. 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: Male Female Total Employed Unemployed 460 40 140 260 600 300 Total 500 400 900 P (M) = probability that a man is chosen P (E) = probability that the one chosen is employed 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: Male Female Total Employed Unemployed 460 40 140 260 600 300 Total 500 400 900 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 a) Arrives on time given that it departed on time b) Departs on time given that it arrived on time c) Arrives on time given that it departed late 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 a) Arrives on time given that it departed on time b) Departs on time given that it arrived on time c) Arrived on time given that it departed late 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 a) Arrives on time given that it departed on time b) Departs on time given that it arrived on time c) Arrived on time given that it departed late 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 a) Arrives on time given that it departed on time b) Departs on time given that it arrived on time c) Arrived on time given that it departed late 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. 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. 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 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 Probability and Statistics Multiplicative Rules Theorem 2.13 If in an experiment the events A and B can both occur, then 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 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? 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? 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. 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 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. 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. 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, Probability and Statistics Theorem 2.16: Total Probability Example 3.20: In a certain plant, three machines, B1, B2, and B3, make 30%, 45%, and 25%, respectively, of the products. It is known from past experience that 2%, 3%, and 2% of the products made by each machine, respectively, are defective. Now, suppose that a finished product is randomly selected. What is the probability that it is defective? A – the product is defective B1 – the product is made by machine B1 B2 – the product is made by machine B2 B3 – the product is made by machine B3 Probability and Statistics Theorem 2.16: Total Probability Example 3.20: In a certain plant, three machines, B1, B2, and B3, make 30%, 45%, and 25%, respectively, of the products. It is known from past experience that 2%, 3%, and 2% of the products made by each machine, respectively, are defective. Now, suppose that a finished product is randomly selected. What is the probability that it is defective? Probability and Statistics Theorem 2.17: Baye’s Rule 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 such that P(A) ) 0 , Probability and Statistics Theorem 2.17: Baye’s Rule Example 3.21: from example 3.20 - if a product were chosen at random and found to be defective, what is the probability that it was made by machine B3? Probability and Statistics Theorem 2.16: Total Probability Example 3.22: Police plan to enforce speed limits by using radar traps at 4 different locations within the city limits. The radar traps at each locations L1, L2 , L3 and L4 are operated 40%, 30%, 20%, and 30% of the time, and if a person who is speeding on his way to work has probabilities of 0.2, 0.1, 0.5 and 0.2, respectively, of passing through these locations, a. What is the probability that he will receive a speeding ticket? b. If the person received a speeding ticket on his way to work, what is the probability that he passed through the radar trap located at L2?