Download Chapter 7: Random Variables

Ch. 5 Review
Algebra III/Stats
Use the following to answer questions 1 and 2:
A psychologist studied the number of sudoku puzzles high school students were able to solve in
a 20-minute period while listening to Mozart. Let X be the number of puzzles completed
successfully by a student. X was found to have the following probability distribution:
Value of X
Probability
1
0.35
2
0.25
3
0.2
4
0.1
5
0.05
6
0.05
1. Referring to the information above, the probability that a randomly chosen subject
completes at least four puzzles in the 20-minute period while listening to soothing
music is
A) 0.2. B) 0.3. C) 0.4. D) 0.9. E) none of these.
2. Referring to the information above, P(X < 3) has value
A) 0.3. B) 0.4. C) 0.6. D) 0.8. E) none of these.
Use the following to answer questions 3 through 5:
Let the random variable X be a randomly generated number with the uniform probability density
curve given below.
3. Referring to the information above, P(X  0) has value
A) 1. B) 0.5. C) 0.1. D) 0.
E) The value cannot be determined since X must be greater than 0.
4. Referring to the information above, P(X = 0.35) is
A) 0.65. B) 0.35. C) 0.05. D) 0.001. E) 0.
5. Referring to the information above, P(0.6 < X < 1.3) has value
A) 0.30. B) 0.40. C) 0.60. D) 0.70. E) none of these.
6. Suppose there are three balls in a box. On one of the balls is the number 1, on another is
the number 2, and on the third is the number 3. You select two balls at random and
without replacement from the box and note the two numbers observed. The sample
space S consists of the three equally likely outcomes {(1, 2), (1, 3), (2, 3)}. X, the sum
of the numbers on the two balls selected, has the following probability distribution:
X
3
4
5
Probability
1/3
1/3
1/3
The probability that X is at least 4 is
A) 0. B) 1/3. C) 9/20. D) 2/3.
E) 1.
Use the following to answer questions 7 through 9:
The probability density curve of a random variable X is given in the figure below.
7. Referring to the information above, the probability that X is between 1.0 and 3.75 is
A) 1/4. B) 1/3. C) 1/2. D) 3/4. E) 2.75.
8. Referring to the information above, the probability that X is at least 0.5 is
A) 0. B) 1/4. C) 1/3. D) 1/2. E) 3/4.
9. Referring to the information above, the probability that X = 1.5 is
A) 0. B) 1/4. C) 1/3. D) 1/2. E) 3/4.
10. If X is binomial with parameters n = 30 and p = 2/3, the mean X of X is
A) 0.6667. B) 4.47. C) 20. D) 30. E) none of these.
11. If X is binomial with parameters n = 30 and p = 2/3, the standard deviation X of X is
A) 6.67 B) 4.47. C) 2.582. D) 1.414. E) 0.6667.
12. In a certain game of chance, your chances of winning are 0.2. If you play the game ten
times and outcomes are independent, the probability that you win at least twice is about
A) 0.0819. B) 0.2. C) 0.3277. D) 0.4096. E) 0.6242.
13. In a certain game of chance, your chances of winning are 0.4. If you play the game six
times and outcomes are independent, the probability that you win all six times is about
A) 1. B) 0.6723. C) 0.3277. D) 0.0041. E) less than 0.0001.
14. In a certain game of chance, your chances of winning are 0.2. You play the game five
times and outcomes are independent. Suppose it costs $1 to play the game each time.
Each time you win, you receive $4 (for a net gain of $3). Each time you lose, you
receive nothing (for a net loss of $1). Your expected winnings for five plays are
A) $3. B) $2. C) $1. D) $0. E) –$1.
Use the following to answer questions 15 through 17:
A survey asks a random sample of 2000 adults in Alabama if they support an increase in the state
sales tax from 5% to 6%, with the additional revenue going to education. Let X denote the
number in the sample that say they support the increase. Suppose that 40% of all adults in
Alabama support the increase.
15. The mean (expected value) of X is
A) 5%.
B) 6%. C) 0.75. D) 40.
16. The standard deviation of X is
A) 800. B) 40. C) 18.97.
E) 800.
D) 21.91.
E) 0.40.
17. The probability that X is more than 900 is about
A) 0.1056. B) 0.2. C) 0.3372. D) 0.4602.
E) none of these.
18. A fair coin (one for which both the probability of heads and the probability of tails are
0.5) is tossed six times. The probability that less than 1/3 of the tosses are tails is about
A) 0.33. B) 0.2061. C) 0.1094. D) 0.09. E) 0.0043.
19. A fair coin (one for which both the probability of heads and the probability of tails are
0.5) is tossed 60 times. The probability that less than 1/4 of the tosses are tails is about
A) 0.33. B) 0.1922. C) 0.0975. D) 0.0875. E) none of these.
20. In a test of ESP (extrasensory perception), the experimenter looks at cards that are
hidden from the subject. Each card contains a star, a circle, a wavy line, or a square. An
experimenter looks at each of 100 cards in turn, and the subject tries to read the
experimenter’s mind and name the shape on each card. What is the probability that the
subject gets at least 30 correct if the subject does not have ESP and is just guessing?
A) 0.1495. B) 0.5000. C) 0.8505. D) 0.9750. E) 0.9999.
21. In a test of ESP (extrasensory perception), the experimenter looks at cards that are
hidden from the subject. Each card contains a star, a circle, a wavy line, or a square. An
experimenter looks at each of 100 cards in turn, and the subject tries to read the
experimenter’s mind and name the shape on each card. What is the probability that the
subject gets at most 30 correct if the subject does not have ESP and is just guessing?
A) 0.9939. B) 0.8962. C) 0.1038. D) 0.0061. E) Less than 0.0001.
22. A multiple-choice exam has 100 questions, each with five possible answers. If a student
is just guessing at all the answers, the probability that he or she gets more than 24
correct is about
A) 0.8686. B) 0.2055. C) 0.1787. D) 0.1314. E) none of these.
23. Suppose X is binomial with parameters n = 9 and p = 1/5. The probability that X is at
most 1 is about
A) 0.5638. B) 0.4362. C) 0.1431. D) 0.1111. E) less than 0.0001.
24. Suppose X is binomial with parameters n = 20 and p = 2/3. The probability that X is at
least 10 is about
A) 0.9624. B) 0.4444. C) 0.3512. D) 0.2634. E) 0.0376.
25. Marcus makes 85% of his free throws. At the end of a game, his team is losing by one
point. He is fouled attempting a three-point shot and is awarded three free throws.
Assuming free throw attempts are independent, what is the probability that he can get
his team the lead (make at least two of the free throws)?
A) 0.50505. B) 0.7. C) 0.84848. D) 0.93925. E) 0.99999.
26. Suppose X has a geometric distribution with probability 0.3 of success and 0.7 of failure
on each observation. The probability that X is equal to 6 is about
A) 0.0081. B) 0.0189. C) 0.0504. D) 0.1029. E) 0.21.
27. Nycole is a lifetime 59% free throw shooter. Suppose this probability is the same for
each free throw she attempts, and free throw attempts are independent. The probability
that she doesn’t make a free throw until her fifth attempt this season is about
A) 0.32769. B) 0.08192. C) 0.0167. D) 0.00032. E) 0.0001.
28. Travis is a lifetime 75% free throw shooter. Suppose this probability is the same for
each free throw he attempts, and free throw attempts are independent. The probability
that he makes his first three free throws and then misses his fourth attempt is
A) 0.4219. B) 0.1055. C) 0.06554. D) 0.0156. E) 0.00032.
29. In a test of ESP (extrasensory perception), the experimenter looks at cards that are
hidden from the subject. Each card contains a star, a circle, some wavy lines, or a
square. The subject tries to read the experimenter’s mind and name the shape on each
card. What is the probability that the subject gets the first four correct before giving a
wrong answer on the fifth try if he is just guessing?
A) 0.00293. B) 0.00391. C) 0.06328. D) 0.07910. E) 0.31641.
30. Holly makes 98% of her free throws. Suppose this probability is the same for each free
throw she attempts, and free throw attempts are independent. The expected number of
free throws required until she makes her first free throw of the season is about
A) 2. B) 1.5. C) 1.02. D) 0.98. E) 0.02
31. In a test of ESP (extrasensory perception), the experimenter looks at cards that are
hidden from the subject. Each card contains a star, a circle, a wavy line, or a square. The
subject tries to read the experimenter’s mind and name the shape on each card. If the
subject is just guessing, what is the expected number of guesses before the subject gets
his first correct guess?
A) 0.25. B) 1.25. C) 2. D) 3. E) none of these.
32. Suppose X has a geometric distribution with probability 0.15 of success and 0.85 of
failure on each observation. The mean and standard deviation of X are
A) mean = .15, standard dev = 37.78.
D) mean = 6.67, standard dev = 37.78.
B) mean = 15, standard dev = 0.85.
E) mean = 6.67, standard dev = 6.15.
C) mean = 2.58, standard dev = 6.15.
33. Amber makes 70% of her free throws. Suppose this probability is the same for each free
throw she attempts, and free throw attempts are independent. The probability that it
takes more than two free throws before she makes her first free throw is about
A) 0.70.
B) 0.50.
C) 0.35.
D) 0.09.
E) cannot be determined from the information given.
34. Suppose X has a geometric distribution with probability 0.3 of success and 0.7 of failure
on each observation. The mean and variance of X are
A) mean = 3.33, variance = 7.78.
D) mean = 0.3, variance = 0.21.
B) mean = 1.43, variance = 0.78.
E) mean = 1.43, variance = 0.61.
C) mean = 3.33, variance = 2.79.
35. A multiple-choice Algebra III/Statistics exam has 50 questions, each with four possible
answer choices. If a student decides to just guess on every question, the probability that
he or she gets less than 10 correct is about
A) 0.8363. B) 0.2000. C) 0.1637. D) 0.0001. E) none of these.
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