A B A or B One or the other or both occurs At least one of A or B occurs Probability Review A B A and B Both A and B occur ( ) P A B : Probability of A given B. Probability that A happens given that B is true or has already happened. Any Events: P A B = P A+ P B P A B ( ) ( ) ( ) ( ) ( ) = ( ) ( ) *I like to think of this as P( A, then B) = P( A) P( B after A) P( A B) # of outcomes in A B ( ) = = P( B) # of outcomes in B P( A B) # of outcomes in A B ( ) = = P( A) # of outcomes in A ( At least one) = 1 P( none) P A B P A P B A P A B P B A P Mutually Exclusive Events (Can t happen at the same time. If one occurs, other is impossible.): P A B = P A+ P B ( ) ( ) ( ) P( A B ) = 0 P( A B ) = 0 P( B A= ) 0 Independent Events (Knowing one has occurred does not change the probability that the other one will.): P A B = P A P B ( ) ( ) ( ) ( ) = P( A) ( ) = P( B) P A B P B A Random Variables: E X µ x p ( ) X i i = ( ) 2 = = σ x µ p 2 X i X i
Binomial setting: Outcomes of each trial can be divided into successes and failures. There are a fixed number of trials. (n) Trials are independent. Knowing the outcome of one gives no information about the outcome of the others. (If sampling without replacement, 10% condition is met). The probability of success is the same for each trial. (p) You are interested in the number of successes. (k) n P( X= k) = p q k ( k) k n k P X= : binompdf (n, p, k): Probability of exactly k successes in n trials. P( X k): binomcdf (n, p, k): Probability of k or fewer successes in n trials. ( ) X E X σ X = = µ = np npq Geometric setting: Outcomes of each trial can be divided into successes and failures The probability of success is the same for each trial. (p) Trials are independent. Knowing the outcome of one gives no information about the outcome of the others. (If sampling without replacement, 10% condition is met). You are interested in when the first success occurs. (k) ( k) ( k) P X= : geometpdf (p, k): Probability that the first success occurs on the kth trial. P X : geometcdf (p, k): Probability that the first success occurs on or before the kth trial. ( ) E X = µ X = 1 p On a free response question, make sure the word binomial or geometric appears in your answer and that all P X= 2 are a plus, but not numbers in the calculator commands are labeled. Probability statements like ( ) required.
1. A weighted die comes up spots with the following probabilities: Spots 1 2 3 4 5 6 Probability 0.1 0.15 0.2 0.25 0.2 0.1 If two of these dice are thrown, what is the probability the sum is 10? A. ( 0.25)( 0.1) + ( 0.2) 2 B. 2( 0.25)( 0.1) + ( 0.2) 2 C. 2( 0.25)( 0.1) + 2( 0.2) 2 D. ( 0.1)( 0.2)( 0.1) + ( 0.2) 2 E. ( 0.1)( 0.2)( 0.1) + ( 0.15)( 0.25) 2 + ( 0.15)( 0.2)( 0.2) + ( 0.25)( 0.1) 2. According to a CBS/New York Times poll taken in 1992, 15% of the public have responded to a telephone call-in poll. In a random group of five people, what is the probability that exactly two have responded to a call-in poll? A. 0.138 B. 0.165 C. 0.300 D. 0.835 E. 0.973 3. The yearly mortality rate for American men from prostate cancer has been constant for decades at about 25 of every 100,000 men. In a group of 100 American men, what is the probability that at least 1 will die from prostate cancer in a given year? A. 0.00025 B. 0.0247 C. 0.025 D. 0.9753 E. 0.99975 4. Suppose that among the 6000 students at a high school, 1500 are taking honors courses and 1800 prefer watching basketball to watching football. If taking honors courses and preferring basketball are independent, how many students are both taking honors courses and prefer basketball to football? A. 300 B. 330 C. 450 D. 825 E. There is not enough information to answer this question. 5. An inspection procedure at a manufacturing plant involves picking three items at random and then accepting the whole lot if at least two of the three items are in perfect condition. If in reality 90% of the whole lot are perfect, what is the probability that the lot will be accepted? A. 0.600 B. 0.667 C. 0.729 D. 0.810 E. 0.972
6. Suppose that, in a certain part of the world, in any 50-year period the probability of a major plague is 0.39, the probability of a major famine is 0.52, and the probability of both a plague and a famine is 0.15. What is the probability of a famine given that there is a plague? A. 0.240 B. 0.288 C. 0.370 D. 0.385 E. 0.760 7. Given the probabilities P( A ) = 0.4 and P( A B) = 0.6, what is the probability ( ) mutually exclusive? If A and B are independent? A. 0.2, 0.4 B. 0.2, 0.33 C. 0.33, 0.2 D. 0.6, 0.33 E. 0.6, 0.4 P B if A and B are 8. You can choose one of three boxes. Box A has four $5 bills and a single $100 bill, box B has 400 $5 bills and 100 $100 bills, and box C has 24 $1 bills. You can have all of box C or blindly pick one bill out of either box A or box B. Which offers the greatest expected winning? A. Box A B. Box B C. Box C D. Either A or B, but not C E. All offer the same expected winning. 9. Given that 52% of the U.S. population are female and 15% are older than age 65, can we conclude that (0.52)(0.15) = 7.8% are women older than age 65? A. Yes, by the multiplication rule. B. Yes, by conditional probabilities. C. Yes, by the law of large numbers. D. Impossible to tell because we don t know if the events are independent. E. Impossible to tell because we don t know if the events are mutually exclusive. 10. Consider the following table of ages of U.S. senators: Age (yrs.) < 40 40 49 50 59 60 69 70 79 > 79 # of Senators 5 30 36 22 5 2 What is the probability that a senator is under 70 years old given that he or she is at least 50 years old A. 0.580 B. 0.624 C. 0.643 D. 0.892 E. 0.969
Questions 11 15 refer to the following study: One thousand students at a city high school were classified both according to GPA and whether or not they consistently skipped classes. Many skipped classes Few skipped classes GPA < 2.0 2.0 3.0 > 3.0 80 25 5 110 175 450 265 890 255 475 270 1000 11. What is the probability that a randomly-selected student has a GPA between 2.0 and 3.0? A. 0.025 B. 0.227 C. 0.450 D. 0.475 E. 0.506 12. What is the probability that a randomly-selected student has a GPA under 2.0 and has skipped many classes? A. 0.080 B. 0.281 C. 0.285 D. 0.314 E. 0.727 13. What is the probability that a randomly-selected student has a GPA under 2.0 or has skipped many classes? A. 0.080 B. 0.281 C. 0.285 D. 0.314 E. 0.727 14. What is the probability that a randomly-selected student has a GPA under 2.0 given that he or she has skipped many classes? A. 0.080 B. 0.281 C. 0.285 D. 0.314 E. 0.727 15. Are GPA between 2.0 and 3.0 and skipped few classes independent? A. No, because 0.475 0.506 B. No, because 0.475 0.890 C. No, because 0.450 0.475 D. Yes, because of conditional probabilities. E. Yes, because of the product rule.
Questions 16 19 refer to the following study: Five hundred people used a home test for HIV, and then all underwent more conclusive hospital testing. The results are shown in the following table. HIV Healthy Positive Test 35 25 60 Negative Test 5 435 440 40 460 500 16. What is the predictive value of the test? That is, what is the probability that a person has HIV and tests positive? A. 0.070 B. 0.130 C. 0.538 D. 0.583 E. 0.875 17. What is the false-positive rate? That is, what is the probability of testing positive given that the person does not have HIV? A. 0.054 B. 0.050 C. 0.130 D. 0.417 E. 0.875 18. What is the sensitivity of the test? That is, what is the probability of testing positive given that the person has HIV? A. 0.070 B. 0.130 C. 0.538 D. 0.583 E. 0.875 19. What is the specificity of the test? That is, what is the probability of testing negative given that the person does not have HIV? A. 0.125 B. 0.583 C. 0.870 D. 0.950 E. 0.946 20. Suppose that the probability that you will receive an A in AP Statistics is 0.35, the probability that you will receive A s in both AP Statistics and AP Biology is 0.19, and the probability that you will receive an A in AP Statistics but not in AP Biology is 0.17. Which of the following is a proper conclusion? A. The probability that you will receive an A in AP Biology is 0.36 B. The probability that you didn t take AP Biology is 0.01 C. The probability that you will receive an A in AP Biology but not in AP Statistics is 0.18 D. The given probabilities are impossible. E. None of the above.
21. Suppose that 2% of a clinic s patients are known to have cancer. A blood test is developed that is positive in 98% of patients with cancer but is also positive in 3% of patients who do not have cancer. If a person who is chosen at random from the clinic s patients is given the test and it comes out positive, what is the probability that the person actually has cancer? A. 0.02 B. 0.40 C. 0.50 D. 0.60 E. 0.98 22. Sixty-five percent of all divorce cases cite incompatibility as the underlying reason. If four couples file for a divorce, what is the probability that exactly two will state incompatibility as the reason? A. 0.104 B. 0.207 C. 0.254 D. 0.311 E. 0.423 23. Suppose we have a random variable X where the probability associated with the value k is 10 k ( 0.37 ) ( 0.63 ) 10 k for k = 0,..., 10. What is the mean of X? k A. 0.37 B. 0.63 C. 3.7 D. 6.3 E. None of the above 24. A computer technician notes that 40% of computers fail because of the hard drive, 25% because of the monitor, 20% because of a disk drive, and 15% because of the microprocessor. If the problem is not in the monitor, what is the probability that it is in the hard drive? A. 0.150 B. 0.400 C. 0.417 D. 0.533 E. 0.650 25. Suppose that 60% of students who take the AP Statistics exam score 4 or 5, 25% score 3, and the rest score 1 or 2. Suppose further that 95% of those scoring 4 or 5 receive college credit, 50% of those scoring 3 receive such credit, and 4% of those scoring 1 or 2 receive credit. If a student who is chosen at random from among those taking the exam receives college credit, what is the probability that she received a 3 on the exam? A. 0.125 B. 0.178 C. 0.701 D. 0.813 E. 0.822
1. The probability that a person recovers from a particular type of cancer operation is 0.7. Suppose 8 people have the operation. What is the probability that a) exactly 5 recover? b) they all recover? c) at least one of them recovers? d) fewer than half recover? e) more than 6 recover? 2. In a certain dice game, a player repeatedly rolls two six-sided dice until he or she rolls doubles. What is the probability that a) the first doubles is on the 3 rd roll? b) it takes fewer than 5 rolls to roll doubles? c) it takes more than 7 rolls to roll doubles? 3. In an AP Statistics class, 60% of the students are male. Of the males, 12% are left-handed, while 10% of the females are left-handed a) Draw a probability tree of the situation. b) What is the probability that a randomly-selected student is left-handed? c) What is the probability that a randomly-selected student is left-handed and male? d) If a randomly-selected student is left-handed, what is the probability that the student is male?
Answers 1. B 2. A 3. B 4. C 5. E 6. D 7. B 8. E 9. D 10. D 11. D 12. A 13. C 14. E 15. A 16. A 17. A 18. E 19. E 20. D 21. B 22. D 23. C 24. D 25. B 1. a. 0.2541 b. 0.0576 c. 0.9999 d. 0.0580 e. 0.2553 2. a. 0.1157 b. 0.5177 c. 0.2791 3. a. b. 0.112 c. 0.072 d. 0.6429