AP Statistics – Probability Review
1. You roll two fair dice, a green one and a red one.
a) Find P(5 on green and 3 on red)
b) Find P(5 on green and 3 on red) or (3 on green and 5 on red)
2. Two fair dice are thrown, a green one and a red one.
a) What is the probability of getting a sum of 6?
b) What is the probability of getting a sum of 4?
c) What is the probability of getting a sum of 6 or 4?
3. You draw two cards from a standard deck of 52 cards without replacing the first one before drawing the second.
a) Find P(ace on first card and king on second card).
b) Find the probability of drawing an ace and a king in either order.
4. You draw a card from a standard deck of 52.
a) What is the probability that the card is a diamond?
b) What is the probability that the card is the queen of diamonds?
c) What is the probability that the card is a queen or a diamond?
5. June wants to become a policewoman. She must take the physical exam and then the written one. Records of past cadets indicate that the probability of passing the physical exam is 0.82. Then, the probability that a cadet passes the written exam given he or she has passed the physical exam is 0.58. What is the probability that June will pass both exams?
6. A new grading policy has been proposed by the Dean of the College of Education for all education majors. All faculty and students in education were asked to give their opinion about the new grading policy. The results are shown below:
/ Favor / Neutral / Oppose / Row TotalStudents / 353 / 75 / 191 / 619
Faculty / 11 / 5 / 18 / 34
Column Total / 364 / 80 / 209 / 653
Suppose someone at random is selected from the School of Education (either student or faculty). Let S = student, F = faculty, Fa = favor, N = neutral, and O = oppose. Find the following:
a) P(Fa)
b) P(Fa | F)
c) P(Fa | S)
d) P(FFA)
e) Are the events faculty and favor policy independent? Explain or justify.
f) Find P(S | O)
g) Find P(S | Fa)
h) Are the events students and favor policy independent? Explain or justify.
i) Are the events students and favor policy mutually exclusive? Explain or justify.
7. The table below shows how 558 people applying for a credit card were classified according to home ownership and length of time in present job.
Length of Time in Present Job
Home Status / Less than 2 Years / 2 or More Years / Row TotalOwner / 73 / 194 / 267
Renter / 210 / 81 / 291
Column Total / 283 / 275 / 558
Suppose a person is chosen at random from the 558 applicants. Let O = the event the person owns a home, R = the event the person rents, L = event this person has had present job less than two years, and
M = the event the person has had present job tow years or more. Compute the following:
a) P(O)
b) P(O | L)
c) P(O | M)
d) P(OM)
e) P(OM)
f) P(R | L)
g) Are the events O and M independent? Explain/justify.
h) Are the events O and M mutually exclusive? Explain/justify.
8. A private college report contains these statistics:
70% of incoming freshmen attended public schools.
75% of public school students who enroll as freshmen eventually graduate.
90% of other freshmen eventually graduate.
a) Is there any evidence that a freshmen’s chances to graduate may depend upon what kind of high school the student attended? Explain.
b) What percent of freshmen eventually graduate?
9. What percent of students who graduate from the college in exercise 8 attended public high school?
10. A company’s records indicate that on any given day about 1% of their day shift employees and 2% of the night shift employees will miss work. Sixty percent of the employees work the day shift.
a) Is absenteeism independent of shift worked? Explain.
b) What percent of employees are absent on any given day?
c) What percent of the absent employees are on the night shift?
11. Suppose that 23% of adults smoke cigarettes. It’s known that 57% of smokers and 13% of nonsmokers develop a certain lung condition by age 60.
a) Explain how these statistics indicate that lung condition and smoking are not independent.
b) What’s the probability that a randomly selected 60-year-old has this lung condition?
c) What’s the probability that someone with the lung condition was a smoker?
12. Let P(A) = 0.6, P(B) = 0.3, and P(AB) = 0.2. Find:
a) P(AB)
b) P(B’)
c) P(AB’)
13. A coin is tossed three times. Make a tree diagram to model the three tosses.
a) Find the probability of obtaining three tails.
b) Find the probability of obtaining at least one head.
14. In a group of 30 students, 20 hold an Australian passport, 10 hold a Malaysian passport, and 8 hold both passports. The other students hold neither type of passport. A student is selected at random.
a) Draw the Venn Diagram that shows this.
b) Find the probability that a student hold both passports.
c) Find the probability that the student holds neither passport.
d) Find the probability that the student holds only one passport.
15. Urn A contains 9 cubes of which 4 are red. Urn B contains 5 cubes of which 2 are red. A cube is drawn at random and in succession from each urn.
a) Find the probability that both cubes are red.
b) Find the probability that only one cube is red.
c) If only one cube is red, find the probability that it came from urn A.
16. The probability that an animal will still be alive in 12 years is 0.55 and the probability that its mate will still be alive in 12 years is 0.60. Find the probability that:
a) Both will still be alive in 12 years.
b) Only the mate will still be alive in 12 years.
c) At least one of them will still be alive in 12 years.
d) The mate is still alive in 12 years given that only one is sill alive in 12 years.
17. Tony has a 90% chance of passing his math test, while Tanya has an 85% chance of passing the same test. If they both sit for the test, find the probability that:
a) Only one of them passes.
b) At least one of them passes the test.
c) Tanya passed given that at least one passed.
18. The probability that Rory finishes a race is 0.55 and the probability that Millicent finishes the same race is 0.6. Because of team spirit, there is an 80% chance that Millicent will finish the race if Rory finishes the race. Find the probability that:
a) Both will finish the race.
b) Rory finishes the race given that Millicent finishes.
19. A student runs the 100 m, 200 m, and 400 m races at the school athletics day. He has an 80% chance of winning any one given race. Find the probability that he will:
a) Win all 3 races.
b) Win the first and last race only.
c) Win the second race given that he wins at least two races.
20. In a group of 70 people, 42 have fair hair, 34 do not have blue eyes and 23 have fair hair but do not have blue eyes.
a) Find the number of students that have both blue eyes and fair hair.
b) Find the probability that a randomly chosen student has neither blue eyes nor fair hair.
c) If a person with blue eyes is chosen at random from the group, what is the probability that this person has fair hair?
d) Is the characteristic of having blue eyes independent of the characteristic of having fair hair? Explain.
21. Tourist pamphlets indicate that 68% of hotels have swimming pools, 23% have workout facilities, and 19% have both features. What is the probability that a hotel has:
a) A pool or workout facilities?
b) Neither a pool or workout facilities?
c) Workout facilities, but no pool?
22. Employment data indicates that 76% of people invest, 39% of people are married, and that half those that invest are married. What’s the probability that a randomly chosen worker
a) Is neither invests nor is married?
b) Is married, but doesn’t invest?
c) Is married or invests?
23. You draw a card at random from a standard deck of 52 cards. Find each of the following conditional probabilities:
a) The card is a spade, given that it is black.
b) The card is black, given that it is a club.
c) The card is a face card, given that it is red.
d) The card is a Jack, given that it is a face card.
24. Eighty percent of kids who visit a doctor have a fever, and 20% of kids with a fever have body aches. What’s the probability that a kid who goes to the doctor has a fever and body aches?
25. You are dealt a hand of three cards, one at a time. Find the probability of each of the following.
a) The first spade you get is the third card dealt.
b) Your cards are all black.
c) You get no face cards.
d) You have at least one Jack.
e) You get no 8’s.
f) You get all clubs.
g) You have at least one heart.
26. 52% of all American workers have a workplace retirement plan, 64% have health insurance, and 46% have both benefits. We select a worker at random.
a) What’s the probability he has neither employer-sponsored health insurance nor a retirement plan?
b) What’s the probability he has health insurance if he has a retirement plan?
c) Are having health insurance and a retirement plan independent events? Explain.
d) Are having these two benefits mutually exclusive? Explain.
27. If you draw a card at random from a well-shuffled deck, is getting an ace independent of the suit? Explain.