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Elementary Probability Theory

Elementary Probability Theory. 4. Section 4.2. Some Probability Rules—Compound Events. Focus Points. Compute probabilities of general compound events. Compute probabilities involving independent events or mutually exclusive events. Use survey results to compute conditional probabilities.

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Elementary Probability Theory

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  1. Elementary Probability Theory • 4

  2. Section • 4.2 • Some Probability Rules—Compound Events

  3. Focus Points • Compute probabilities of general compound events. • Compute probabilities involving independent events or mutually exclusive events. • Use survey results to compute conditional probabilities.

  4. 1. Conditional Probability and Multiplication Rules

  5. 1. Conditional Probability and Multiplication Rules • Independent events • Two events are independent if the occurrence or nonoccurrence of one does not change the probability that the other will occur. • Multiplication rule for independent events: • P(A and B) = P(A)  P(B) (4)

  6. 1. Conditional Probability and Multiplication Rules • If two events are dependent, then we must take intoaccount the changes in the probability of one event caused by the occurrence of the other event. • The notation P(A, given B) denotes the probability that event A will occur given that event B has occurred. • This is called a conditional probability.

  7. 1. Conditional Probability and Multiplication Rules • We read P(A, given B) as “probability of A given B.” If A and B are dependent events, then P(A) P(A, given B) because the occurrence of event B has changed the probability that event A will occur. • A standard notation for P(A, given B) is P(A|B). • General multiplication rule for any events: • P(A and B) = P(A) P(B|A) (5) • P(A and B) = P(B) P(A|B) (6)

  8. 1. Conditional Probability and Multiplication Rules • We will use either formula (5) or formula (6) according to the information available. • Formulas (4), (5), and (6) constitute the multiplication rules of probability. • They help us compute the probability of events happening together when the sample space is too large for convenient reference or when it is not completely known.

  9. 1. Conditional Probability and Multiplication Rules • Note: For conditional probability, observe that the multiplication rule • P(A and B) = P(B) P(A|B) • can be solved for P(A|B), leading to

  10. Practice 1: Independent events • A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die. • Solution: • P(head and 4) = P(head)  P(4) = ½ ⅙ =1/12 • Referece: Bluman, Elementary Statistics, 2e Page 206.

  11. Practice 2: Independent events • A card is drawn from a deck and replaced: then a second card is drawn. Find the probability of getting a queen and then an ace. • Solution: • P(queen and ace) = P(queen)  P(ace) • = (4/52)  (4/52 = 1/169 • Referece: Bluman, Elementary Statistics, 2e Page 206.

  12. Example 4 – Multiplication Rule, Independent Events • Suppose you are going to throw two fair dice. What is the probability of getting a 5 on each die? • Solution Using the Multiplication Rule: • The two events are independent, so we should use formula (4). • P(5 on 1st die and 5 on 2nd die) • = P(5 on 1st) P(5 on 2nd) • To finish the problem, we need to compute the probability of getting a 5 when we throw one die.

  13. Example 4 – Solution • There are six faces on a die, and on a fair die each is equally likely to come up when you throw the die. Only one face has five dots, so by formula (2) for equally likely outcomes, • Now we can complete the calculation. • P(5 on 1st die and 5 on 2nd die) = P(5 on 1st) P(5 on 2nd)

  14. Example 4 – Solution • cont’d • Solution Using Sample Space: • The first task is to write down the sample space. Each die has six equally likely outcomes, and each outcome of the second die can be paired with each of the first. The sample space is shown in Figure 4-2. • Sample Space for Two Dice • Figure 4-2

  15. Example 4 – Solution • cont’d • The total number of outcomes is 36, and only one is favorable to a 5 on the first die and a 5 on the second. The 36 outcomes are equally likely, so by formula (2) for equally likely outcomes, • P(5 on 1st and 5 on 2nd) = • The two methods yield the same result. The multiplication rule was easier to use because we did not need to look at all 36 outcomes in the sample space for tossing two dice.

  16. 1. Conditional Probability and Multiplication Rules • Procedure:

  17. Practice 3: Dependent events • Three cards are drawn from an ordinary deck and not replaces. Find the probability of these:a) Getting three jacksb) Getting an ace, a king, and a queen in orderc) Getting a club, a spade, and a heart in orderd) Getting three clubs • Solution: • a) (4/52)(3/51)(2/50)=1/5525b) (4/52)(4/51)(4/50)=8/16575c) (13/52)(13/51)(13/50)=169/10200d) (13/52)(12/51)(11/51)=11/850 • Referece: Bluman, Elementary Statistics, 2e Page 209.

  18. Practice 4: Conditional Probability • A box contains black chips and white chips. A person selects two chips without replacement. If the probability of selecting a black chip and a white chip is 15/56, and the probability of selecting a black chip on the first draw is 3/8, find the probability of selecting the white chip on the second draw , given that the first chip selected was a black chip. • Solution: • Let B = selecting a black chip W = selecting a white chipThen • P(W|B) = P(B and W)/P(B) = (15/56) / (3/8) = 5/7 • Referece: Bluman, Elementary Statistics, 2e Page 211.

  19. 2. Addition Rules

  20. 2. Addition Rules • One of the multiplication rules can be used any time we are trying to find the probability of two events happening together. Pictorially, we are looking for the probability of the shaded region in Figure 4-4(a). • The Event A and B • Figure 4-4(a)

  21. 2. Addition Rules • Another way to combine events is to consider the possibility of one event oranother occurring. • For instance, if a sports car saleswoman gets an extra bonus if she sells a convertible or a car with leather upholstery, she is interested in the probability that you will buy a car that is a convertible or that has leather upholstery. • Of course, if you bought a convertible with leather upholstery, that would be fine, too.

  22. 2. Addition Rules • Pictorially, the shaded portion of Figure 4-4(b) represents the outcomes satisfying the orcondition. • The Event A or B • Figure 4-4 (b)

  23. 2. Addition Rules • Notice that the condition A or B is satisfied by any one of the following conditions: • 1. Any outcome in A occurs. • 2. Any outcome in B occurs. • 3. Any outcome in both A and B occurs. • It is important to distinguish between the orcombinations and the andcombinations because we apply different rules to compute their probabilities.

  24. 2. Addition Rules • Once you decide that you are to find the probability of an “or” combination rather than an “and” combination, what formula do you use? • It depends on the situation. In particular, it depends on whether or not the events being combined share any outcomes. • Example 6 illustrates two situations.

  25. Guided Exercise 6: Combining Events • Indicate how each of the following pairs of events are combined. Use either the and combination or the or combination. • Satisfying the humanities requirement by taking a course in the history of Japan or by taking a course in classical literature • Buying new tires and aligning the tires • Getting an A not only in psychology but also in biology • Having at least one of these pets: cat, dog, bird, rabbit

  26. Example 6 – Probability of Events Combined with Or • Consider an introductory statistics class with 31 students. The students range from freshmen through seniors. Some students are male and some are female. Figure 4-5 shows the sample space of the class. • Sample Space for Statistics Class • Figure 4-5

  27. Example 6 – Probability of Events Combined with Or • Suppose we select one student at random from the class. Find the probability that the student is either a freshman or a sophomore. P(freshmen or Sophomore)=15/431+8/31=23/31 ≈0.742

  28. Example 6 – Probability of Events Combined with Or • (b) Select one student at random from the class. What is the probability that the student is either a male or a sophomore? P(male or sophomore)=14/31+8/31-5/31=17/31 ≈0.548

  29. Addition Rules • We say the events A and B are mutually exclusive or disjointif they cannot occur together. This means that A and B have no outcomes in common or, put another way, that P(A and B) = 0. • Formula (7) is the addition rule for mutually exclusive events A and B.

  30. 2. Addition Rules • If the events are not mutually exclusive, we must use the more general formula (8), which is the general addition rule for any events A and B.

  31. 2. Addition Rules • Procedure:

  32. Example 7 – Mutually Exclusive Events • Laura is playing Monopoly. On her next move she needs to throw a sum bigger than 8 on the two dice in order to land on her own property and pass Go. What is the probability that Laura will roll a sum bigger than 8? • Solution: • When two dice are thrown, the largest sum that can come up is 12. Consequently, the only sums larger than 8 are 9, 10, 11, and 12. • These outcomes are mutually exclusive, since only one of these sums can possibly occur on one throw of the dice.

  33. Example 7 – Solution • cont’d • The probability of throwing more than 8 is the same as • P(9 or 10 or 11 or 12) • Since the events are mutually exclusive, • P(9 or 10 or 11 or 12) = P(9) + P(10) + P(11) + P(12)

  34. Example 7 – Solution • cont’d • To get the specific values of P(9), P(10), P(11), and P(12), we used the sample space for throwing two dice (see Figure 4-2). • Sample Space for Two Dice • Figure 4-2

  35. Guided Exercise 8: Addition Rule • Professor Jackson is in charge of a program to prepare people for a high school equivalency exam. Records show that 80% of the students need work in math, 70% need work in English, and 55% need work in both areas. • Are the events needs math and needs English mutually exclusive? • Use the appropriate formula to compute the probability that a student selected at random needs math or English. • Solution: (b)P(needs math or needs English) • = 0.80 + 0.70 – 0.55 = 0.95

  36. Practice 4: Complement event • A coin is tossed five times. Find the probability of getting at least one tail. • Solution: • It is easier to find the probability of the complement of the event, which is “all heads”, and then subtract the probability from 1 to get the probability of at one tail. • P(E) = 1 – P(Ē) • P(at least 1 tail) = 1 – P(all heads) • P(all heads) = 1/32 • P(at least 1 tail)= 1 – 1/32 = 31/32 • Referece: Bluman, Elementary Statistics, 2e Page 213.

  37. Example 8: Survey • At Hopewell Electronics, all 140 employees were asked about their political affiliations. The employees were grouped by type of work, as executives or production workers. The results with row and column totals are shown in Table 4-2.

  38. Example 8: Survey • Compute P(D) and P(E) • Compute P(D|E) • Are the events D and E independent? • Compute P(D and E) • Compute P(D or E)

  39. Answers to some even hw: • 2. 0.3 • 8. a) 111/288 b) 81/288 c) 237/288 d) 159/288 e) 18/288 • 10. a) Yes b) 1/36 c) 1/36 d) 1/18 • 12. a) 1/6 b) 1/18 c) 2/9; Yes • 14. a) No b) 0.006 c) 0.006 c) 0.012 • 20. a) 110/130 b) 20/130 c) 50/70 d) 20/70 e) 110/200 f) 20/200 • 24. a) 291/2008 b) 77/452 c) 826/2008 d) 131/373 e) 41/152 f) 53/157 g) 420/452 h) 332/373 i) No. P(15+ years) = 535/2003 ≠ P(15+ years | east) = 118/452

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