Probability We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions at a faster rate. Use your own judgment, based on the group of students, to determine the order and selection of questions to work in the session. Be sure to include a variety of types of questions (multiple choice and free response) in the time allotted. Multiple Choice 1. A Let A represent an attendee who graduated in 1998 and let B represent an attendee who went 37 into the military. P( A B) 0.072 513 2. B Let A represent an attendee who graduated in 1968 and let B represent an attendee who went 56 to college. P( B A) 0.253 221 3. B 1 5 P(not a five)=1 P(five) 1 6 6 4. E Since mutually exclusive events have no outcomes in common, knowing that one of the events occurred means that the other event does not occur. 5. C Since the normal distribution is symmetric and 55 is one standard deviation above the mean, the proportion of values above 55 would be equal to the proportion of values below one standard deviation less than the mean, (the proportion of values less than 50 5 45 ). 6. E The probability is based on the past outcomes of that particular flight arriving in Los Angeles on time. 7. B Let X represent the number of people that say global climate change is a threat. X is binomial with n 25 and p 0.4. P( X 14) 1 P( X 14) 0.0344 8. E Let X be the number of points earned on one spin. EX ( ) 0.2(15) 0.8( 4) 0.2 Let Y be the outcome of 40 spins. E( Y) 40 E( X ) 40(.2) 8 points 9 D Let X be the number of sweatshirts sold by the vendor. EX ( ) (0)(0.3) (1)(0.2) (2)(0.3) (3)(0.1) (4)(0.08) (5)(0.02) 1.52 sweatshirts Let Z be the total dollar amount taken in by the vendor. E( Z) $25( E( X )) $25(1.52) $38.00 10. D 2 2 2 2 2 2 2 2 X 1.33 Therefore Y 25 X 25... 25 7(25) (1.33) 87.971 1 X 2 X 7
11. E Since there are no parents who voted for beef jerky the events are mutually exclusive. The probability of popcorn or student is: P(popcorn or student) P( p) P( st) P( p st) 0.3333 0.5556 0.1296 0.7593 Free Response 12. Similar to 2004 Q4 Solution Part (a): Let A be the event medicine A works. Let B be the event medicine B works. The probability that a pilot will be cured with Plan I is: P( Cure ) P( A) P( not A) P( B) I 0.4 (0.6)(0.8) 0.88 The probability that a pilot will be cured with Plan II is: P( Cure ) P( B) P( not B) P( A) II 0.8 (0.2)(0.4) 0.88 Part (b): Treatment with medicine A costs $10, and treatment with medicine B costs $40. The expected cost per pilot when Plan I is used for treatment is: P( Cost ) ($10)(0.4) ($50)(0.6) I $4 $30 $34 The expected cost per pilot when Plan II is used for treatment is: P( Cost ) ($40)(0.8) ($50)(0.2) II $32 $10 $42 Part (c): Since the probability that a pilot will be cured is the same under either plan, some other criterion must be used to make a recommendation. From a financial point of view, Plan I should be recommended because the expected cost per pilot is less than Plan II.
Scoring Each part is scored as essentially correct, partially correct, or incorrect. Part (a) is essentially correct if the probabilities of cure are calculated correctly with justification for both plans. Plan I: Plan II: P( CureI ) 0.4 (0.6)(0.8) 0.88 P( CureI ) P( A B) 0.4 0.8 (0.4)(0.8) 0.88 P( Cure ) 1 P( not A) P( not B) 1 (0.6)(0.2) 0.88 I P( CureII ) 0.8 (0.2)(0.4) 0.88 P( CureII ) P( B A) 0.8 0.4 (0.8)(0.4) 0.88 P( Cure ) 1 P( not B) P( not A) 1 (0.2)(0.6) 0.88 II Part (a) is partially correct if one of the two probabilities is calculated correctly with justification, both probabilities are correct with incomplete justifications.
Part (b) is essentially correct if the expected costs per pilot are calculated correctly with justification for both plans. The expected cost per pilot when Plan I is used for treatment is: P( Cost ) ($10)(0.4) ($50)(0.6) P( Cost ) ($10) ($40)(0.6) I $4 $30 $34 I $10 $24 $34 The expected cost per pilot when Plan II is used for treatment is: P( Cost ) ($40)(0.8) ($50)(0.2) P( Cost ) ($40) ($10)(0.2) II $32 $10 $42 II $40 $2 $42 Part (b) is partially correct if the expected cost per pilot is calculated correctly with justification for one of the two plans, both expected costs are correct with incomplete justifications, the expected costs are incorrectly calculated but the probabilities involved add up to 1. Part (c) is essentially correct if the recommendation contains a statistical argument based on parts (a) and (b). That is, the student must base the recommendation on probabilities from part (a) and expected values from part (b). The following two examples are essentially correct: Since the probability that a pilot will be cured is the same under either plan, some other criterion must be used to make a recommendation. From a financial point of view, Plan I should be recommended because the expected cost per pilot is less than Plan II. Since the probability that a pilot will be cured is the same under either plan, some other criterion must be used to make a recommendation. The airline might prefer Plan II, regardless of its higher cost, because the pilot is more likely to need only the first drug. Part (c) is partially correct if the recommendation contains a statistical argument based only on part (a) or (b) but not both. Part (c) is incorrect if no recommendation is made.
4 Complete Response All three parts essentially correct 3 Substantial Response Two parts essentially correct and 1 part partially correct 2 Developing Response Two parts essentially correct and no parts partially correct One part essentially correct and 2 parts partially correct Three parts partially correct 1 Minimal Response One part essentially correct and either 0 or 1 part partially correct No parts essentially correct and 2 parts partially correct
13. Similar to 2005B Q2 Solution Part (a): The mean of A is: 0 0.1 1 0.15 2 0.55 3 0.05 4 0.15 2. The standard deviation of A is: 2 2 2 2 2 (0 2) 0.1 (1 2) 0.15 (2 2) 0.55 (3 2) 0.05 (4 2) 0.15 1.095. Part (b): Let T A C, where C is the total number of children s shirts purchased by a single customer, denote the total number of tickets purchased by a single customer. The mean of T is 2 2 4. T A C 2 2 2 2 The standard deviation of T is 1.095 1.5 3.45 1.857. T A C Part (c): Let M 12 C 24 A, denote the total amount of money spent per purchase. The mean of M is M 12 C 24 A 12 2 24 2 $72. The standard deviation of M is. 2 2 2 2 2 2 M 12 C 24 A 144 1.5 576 1.095 1468.8 $38.32 Scoring Each part is scored as essentially correct (E), partially correct (P), or incorrect (I). Part (a) is essentially correct (E) if both the mean and the standard deviation of A are calculated correctly and work is shown. Part (a) is partially correct (P) if either the mean or the standard deviation of A is calculated correctly and the work is shown. Note: If the variance is reported instead of the standard deviation, the response is scored as (P). Part (a) is incorrect (I) if both the mean and the standard deviation Notes: 1. Unsupported answers will be scored as incorrect. 2. If the student incorrectly calculates the mean and/or standard deviation in part (a) and then correctly uses those values in parts (b) and (c), there will be no second penalty. 2 2 2 2 3. Standard notation for means (,,, and ), variances (,,, and ), and C A T M C A T M standard deviations (,,, and ) are acceptable without definition. If C A T M nonstandard notation such as ( pc, pa, pt, and p M ), is defined correctly for this problem, then it will be scored essentially correct. Nonstandard notation, without a definition, will be scored at most partially correct.
Part (b) is essentially correct (E) if both the mean and the standard deviation of T are calculated correctly and the work is shown, with the exception of minor arithmetic errors. Part (b) is partially correct (P) if either the mean or the standard deviation of T is calculated correctly. Part (b) is incorrect (I) if both the mean and the standard deviation of T are calculated incorrectly no work is shown. Part (c) is scored as essentially correct (E) if both the mean and the standard deviation of M are calculated correctly and the work is shown, with the exception of minor arithmetic errors. Part (c) is partially correct (P) if either the mean or the standard deviation of M is calculated correctly. Part (c) is incorrect (I) if both the mean and the standard deviation of M are calculated incorrectly no work is shown. 4 Complete Response (3E) All three parts essentially correct 3 Substantial Response (2E 1P) Two parts essentially correct and one part partially correct 2 Developing Response (2E 0P or 1E 2P or 3P) Two parts essentially correct and zero parts partially correct One part essentially correct and two parts partially correct All three parts partially correct 1 Minimal Response (1E 1P or 1E 0P or 0E 2P) One part essentially correct and either zero parts or one part partially correct Zero parts essentially correct and two parts partially correct
14. Similar to 2003 Q3 Solution Part (a): P( footsize 24 or footsize 32) P( footsize 24) P( footsize 32) 24 27 32 27 P z P z 1.6 1.6 P( z 1.875) P( z 3.125) 0.0304 0.0009 0.0313 Part (b): P(28 footsize 30) 28 27 32 27 P z 1.6 1.6 P 0.625 z 1.875 0.2356 Part (c): X number of customers who request size L X is binomial with n 8 customers and p 0.2356 PX 8 3 3 5 ( 3) (0.2356) (0.7644) 0.1911
Scoring Each part is scored as essentially correct (E), partially correct (P), or incorrect (I). Part (a) is essentially correct (E) if the response 1. recognizes the need to look at foot lengths below 24 cm and above 32 cm 2. correctly computes the two tail probabilities (except for minor arithmetic or transcription errors) and adds those probabilities Part (a) is partially correct (P) if the response considers only foot lengths below 24 cm (or above 32 cm) but computes the corresponding tail area correctly recognizes the need to look at foot lengths below 24 cm and above 32 cm but does not compute both tail probabilities correctly recognizes the need to look at foot lengths below 24 cm and above 32 cm but approximates tail probabilities using the Empirical Rule computes the proportion of customers that will find that the store carries their size (i.e., 1 correct answer ) States the correct answer (0.0313) without supporting work NOTE: A normal curve with correct regions shaded showing both correct end points (24 and 32) and the mean and standard deviation may be used for element 1. Part (b) is essentially correct (E) if the response 1. the appropriate probability is illustrated using a normal curve in which the end points are identified and the mean and standard deviation are implied 2. the required probability is correctly computed (except for minor arithmetic errors) Part (b) is partially correct (P) if only one of the above elements is correct. NOTES: 1. If part (a) was not essentially correct because the student interchanged the mean and standard deviation, and the same values for mean and standard deviation are used in part (b), the part (b) can be considered essentially correct if the probability calculated is correct for the mean and standard deviation used. 2. A reasonable approximation using the Empirical Rule in part (b) is only acceptable if the computation in part (a) is done correctly (i.e., without using the Empirical Rule).
Part (c) is essentially correct (E) if 1. the student recognized the setting as binomial 2. the probability calculated in part (b) is used for p 3. work is shown that is, the correct values for n and x are given and the desired probability calculated, or the binomial formula is correctly evaluated. Part (c) is partially correct (P) if the student recognizes the situation as binomial and identifies p from part (b) but does not computed the desired probability the student computes the probability as either 3 5 (0.2356) (0.7644) or the student gives the correct probability of 0.1911 but work is not shown NOTE: Rounding the probability in part (b) for use in part (c) is acceptable. 4 Complete Response (3E) All three parts essentially correct 3 Substantial Response (2E 1P) Two parts essentially correct and one part partially correct 2 Developing Response (2E 0P or 1E 2P or 3P) Two parts essentially correct and zero parts partially correct One part essentially correct and two parts partially correct All three parts partially correct 8 (0.2356) 3 3 1 Minimal Response (1E 1P or 1E 0P or 0E 2P) One part essentially correct and either zero parts or one part partially correct Zero parts essentially correct and two parts partially correct