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465_SampleQuiz_2

# 465_SampleQuiz_2

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# Math 465

Sample Quiz #2

Winter 2011

Section 1.4 Exercises (page 36) Problems: 1.4-1 through 1.4-12, 1.4-14, 1.4-16 through 1.4-18, 1.4-19 (a) through (d) Section 1.5 Exercises (page 43) Problems: 1.5-1 through 1.5-17, 1.5-19 (a), 1.5-20 (a) and (b) 1. A box contains 20 red balls, 30 white balls, and 50 blue balls. Suppose that 10 balls are selected at random one at a time, with replacement. Determine the probability that at least one color will be missing from the selected balls. (.5)10 + (.7)10 + (.8)10 − ((.2)10 + (.3)10 + (.5)10 ) 2. Two distinct dice are rolled repeatedly and the sum of the outcomes is observed for each roll. Find the probability that the sum 9 is observed before the sum 7. 4 P(the sum is 9| the sum is 7 or 9) = 6+4 3. Suppose that two players A and B take turns rolling a pair of balanced dice, and that the winner is the ﬁrst player who obtains the sum of 7 on a given roll of the two dice. If A rolls ﬁrst, what is the probability that B will win? 51 5551 555551 15 52 54 5 1 52 6 6 + 6 6 6 6 + 6 6 6 6 6 6 + . . . = 6 6 (1 + 62 + 64 + . . .) = 62
1− 62

4. Three players A, B, and C take turn tossing a fair coin. Suppose that A tosses the coin ﬁrst, B tosses second, and C tosses third; and this cycle is repeated indeﬁnitely until someone wins by being the ﬁrst player to obtain a head. Determine the probability that each of three players will win? 1 1 P(A wins) = 1 + 24 + . . . = 1−2 1 = 4 2 7 P(B wins) = 2 , P(C wins) = 7
1 7
23

5. An urn contains 9 balls marked LOSE and 3 balls marked WIN. You and an opponent take turns selecting at random a single ball from the urn without replacement. The person who selects the third WIN ball wins the game. It does not matter who selected the ﬁrst two WIN balls. (a) If you draw ﬁrst, ﬁnd the probability that you win the game on your second draw. (3) 1 2 (12) 10 2 (b) If you draw ﬁrst, ﬁnd the probability that your opponent wins the game on his second draw. (3)(9) 1 1 2 (12) 9 3 (c) If you draw ﬁrst, ﬁnd the probability that you win the game. Hint: You could win on your second, third, fourth , . . ., or tenth draw, not on your ﬁrst. (3) 1 (3)(9) 1 (3)(9) 1 (3)(9) 1 (3)(9) 1 2 2 2 + 2 12 4 6 + 2 12 6 4 + 2 12 8 2 12 10 + (2) (12) 8 (6) (8) (10) 4 6. In a string of 12 Christmas tree light bulbs, 3 are defective. The bulbs are selected at random and tested, one at a time, until the third defective bulb is found. Compute the probability that the third defective bulb is the (a) fourth bulb tested. (3)(9) 1 2 1 (12) 9 3 (b) ninth bulb tested (3)(9) 1 2 6 (12) 4 8 7. A small grocery store had 12 cartons of milk, 3 of which were sour. If you are going to buy the ﬁfth carton of milk sold that day at random, compute the probability of selecting a carton of sour milk. (3)(9) 1 2 6 (12) 4 8 8. You are a member of a class of 18 students. A bowl contains 18 chips, 3 blue and 15 red. Each student is to take 1 chip from the bowl without replacement. The student who draws the blue chip is guaranteed an A for the course. If you are drawing sixth, ﬁnd the probability that you will 1

+ (−1) n! 12. A match occurs if ball number m is the mth ball selected. and 10 olive socks. 4 P(SUM is 7|SUM is 7 or SUM is 8)= 4+6 .4-19 An urn contains four balls numbered 1 through 4. If you draw ﬁrst.10 (2) (8)+(6)+(10) 2 2 2 10. Bowl A contains 3 red and 2 white chips. 1 − 1 2! 1 3! + − 1 4! 2 . (c) P(Ai ∩ Aj ∩ Ak ) = 1! . . Flip an unbiased coin seven independent times. i = 1. one at a time. . m. Find the probability of at least one match if the sampling is done (a) With replacement. 1. . Show that 2! 3! (a) P(Ai ) = 4! . 7 1 7 3 (2) 14. (3)(33) 3 (3)(33) 2 (3)(33) 1 0 5 + 1 36 4 31 + 2 36 3 31 36 ( 5 ) 31 (5) (5) 9. 2.) = 1 (1 + 5 42 52 + 44 54 + .) = 1 1 5 1− 42 52 = 5 9 (b) Without replacement. . An urn contains four balls numbered 1 through 4. (15)(15) 13 2 3 (30) 25 5 15. 1 ( 2 )7 (b) Three heads occurring in seven trials. (8)+(6)+(10) 2 2 2 (24) 2 (b) Compute the probability that both socks are olive. You and another player take turn selecting a ball from the urn. ﬁnd the probability that the third white ball is the sixth ball drawn if the sampling is done (a) With replacement. Suppose that two dice are to be rolled repeatedly and the sum T of the two numbers is to be observed for each roll. 13. (b) P(Ai ∩ Aj ) = 4! . A match occurs if ball number m is the mth ball selected. . . A drawer contains 8 black. . one marked WIN and four marked LOSE. ﬁnd the probability that you will win if the sampling is done (a) With replacement. . We want to determine the probability that the value T=7 will be observed before the value T=8 is observed. 1 5 + 1 5 441 555 + 44441 55555 + . and bowl B contains 4 red and 3 white chips. The balls are selected one at a time without replacement. one at a time. . 6 brown. 5 1 2 1 31 2 (2) (2) 2 (b) Without replacement. Compute the probability of then drawing red chip from bowl B. The balls are drawn from the urn at random. Show that the probability of at least one match is P(A1 ∪ A2 ∪ . A chip is drawn at random from bowl A and transferred to bowl B. ∪ An ) = n+1 1 1 1 1 − 2! + 3! − 4! + . An urn contains 15 red balls and 15 white balls. if it is known that they are of the same color. Two socks are selected at random from the drawer. 24 35 58 + 58 11. . (d) The probability of at least 4! 1 1 1 one match is P(A1 ∪ A2 ∪ A3 ∪ A4 ) = 1 − 2! + 3! − 4! . Let the event Ai denote a match on the ith draw. 1 − ( 3 )4 4 (b) Without replacement. .draw a blue chip. . + (4) 1 (4) 1 2 + 4 (5) 3 (5) 1 2 4 16. (a) Compute the probability that both socks are of the same color. The ﬁrst person to select WIN ball is the winner. An urn contains ﬁve balls. The balls are selected one at a time. (e) Extend this exercise so that there are n balls in the urn. Compute the probability of (a) HTHTHHT.

scribd
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The text passage is a sample quiz containing several probability and combinatorics problems, each with a question and an answer provided.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
P. 1
465_SampleQuiz_2

# 465_SampleQuiz_2

|Views: 521|Likes:
Published by Anh Thu Tran Xuan

### More info:

Published by: Anh Thu Tran Xuan on Feb 10, 2012
Copyright:Attribution Non-commercial

### Availability:

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05/07/2015

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# Math 465

Sample Quiz #2

Winter 2011

Section 1.4 Exercises (page 36) Problems: 1.4-1 through 1.4-12, 1.4-14, 1.4-16 through 1.4-18, 1.4-19 (a) through (d) Section 1.5 Exercises (page 43) Problems: 1.5-1 through 1.5-17, 1.5-19 (a), 1.5-20 (a) and (b) 1. A box contains 20 red balls, 30 white balls, and 50 blue balls. Suppose that 10 balls are selected at random one at a time, with replacement. Determine the probability that at least one color will be missing from the selected balls. (.5)10 + (.7)10 + (.8)10 − ((.2)10 + (.3)10 + (.5)10 ) 2. Two distinct dice are rolled repeatedly and the sum of the outcomes is observed for each roll. Find the probability that the sum 9 is observed before the sum 7. 4 P(the sum is 9| the sum is 7 or 9) = 6+4 3. Suppose that two players A and B take turns rolling a pair of balanced dice, and that the winner is the ﬁrst player who obtains the sum of 7 on a given roll of the two dice. If A rolls ﬁrst, what is the probability that B will win? 51 5551 555551 15 52 54 5 1 52 6 6 + 6 6 6 6 + 6 6 6 6 6 6 + . . . = 6 6 (1 + 62 + 64 + . . .) = 62
1− 62

4. Three players A, B, and C take turn tossing a fair coin. Suppose that A tosses the coin ﬁrst, B tosses second, and C tosses third; and this cycle is repeated indeﬁnitely until someone wins by being the ﬁrst player to obtain a head. Determine the probability that each of three players will win? 1 1 P(A wins) = 1 + 24 + . . . = 1−2 1 = 4 2 7 P(B wins) = 2 , P(C wins) = 7
1 7
23

5. An urn contains 9 balls marked LOSE and 3 balls marked WIN. You and an opponent take turns selecting at random a single ball from the urn without replacement. The person who selects the third WIN ball wins the game. It does not matter who selected the ﬁrst two WIN balls. (a) If you draw ﬁrst, ﬁnd the probability that you win the game on your second draw. (3) 1 2 (12) 10 2 (b) If you draw ﬁrst, ﬁnd the probability that your opponent wins the game on his second draw. (3)(9) 1 1 2 (12) 9 3 (c) If you draw ﬁrst, ﬁnd the probability that you win the game. Hint: You could win on your second, third, fourth , . . ., or tenth draw, not on your ﬁrst. (3) 1 (3)(9) 1 (3)(9) 1 (3)(9) 1 (3)(9) 1 2 2 2 + 2 12 4 6 + 2 12 6 4 + 2 12 8 2 12 10 + (2) (12) 8 (6) (8) (10) 4 6. In a string of 12 Christmas tree light bulbs, 3 are defective. The bulbs are selected at random and tested, one at a time, until the third defective bulb is found. Compute the probability that the third defective bulb is the (a) fourth bulb tested. (3)(9) 1 2 1 (12) 9 3 (b) ninth bulb tested (3)(9) 1 2 6 (12) 4 8 7. A small grocery store had 12 cartons of milk, 3 of which were sour. If you are going to buy the ﬁfth carton of milk sold that day at random, compute the probability of selecting a carton of sour milk. (3)(9) 1 2 6 (12) 4 8 8. You are a member of a class of 18 students. A bowl contains 18 chips, 3 blue and 15 red. Each student is to take 1 chip from the bowl without replacement. The student who draws the blue chip is guaranteed an A for the course. If you are drawing sixth, ﬁnd the probability that you will 1

+ (−1) n! 12. A match occurs if ball number m is the mth ball selected. and 10 olive socks. 4 P(SUM is 7|SUM is 7 or SUM is 8)= 4+6 .4-19 An urn contains four balls numbered 1 through 4. If you draw ﬁrst.10 (2) (8)+(6)+(10) 2 2 2 10. Bowl A contains 3 red and 2 white chips. 1 − 1 2! 1 3! + − 1 4! 2 . (c) P(Ai ∩ Aj ∩ Ak ) = 1! . . Flip an unbiased coin seven independent times. i = 1. one at a time. . m. Find the probability of at least one match if the sampling is done (a) With replacement. 1. . Show that 2! 3! (a) P(Ai ) = 4! . 7 1 7 3 (2) 14. (3)(33) 3 (3)(33) 2 (3)(33) 1 0 5 + 1 36 4 31 + 2 36 3 31 36 ( 5 ) 31 (5) (5) 9. 2.) = 1 (1 + 5 42 52 + 44 54 + .) = 1 1 5 1− 42 52 = 5 9 (b) Without replacement. . An urn contains four balls numbered 1 through 4. (15)(15) 13 2 3 (30) 25 5 15. 1 ( 2 )7 (b) Three heads occurring in seven trials. (8)+(6)+(10) 2 2 2 (24) 2 (b) Compute the probability that both socks are olive. You and another player take turn selecting a ball from the urn. ﬁnd the probability that the third white ball is the sixth ball drawn if the sampling is done (a) With replacement. Suppose that two dice are to be rolled repeatedly and the sum T of the two numbers is to be observed for each roll. 13. (b) P(Ai ∩ Aj ) = 4! . A match occurs if ball number m is the mth ball selected. . . A drawer contains 8 black. . one marked WIN and four marked LOSE. ﬁnd the probability that you will win if the sampling is done (a) With replacement. . We want to determine the probability that the value T=7 will be observed before the value T=8 is observed. 1 5 + 1 5 441 555 + 44441 55555 + . and bowl B contains 4 red and 3 white chips. The balls are selected one at a time without replacement. one at a time. . 6 brown. 5 1 2 1 31 2 (2) (2) 2 (b) Without replacement. Compute the probability of then drawing red chip from bowl B. The balls are drawn from the urn at random. Show that the probability of at least one match is P(A1 ∪ A2 ∪ . A chip is drawn at random from bowl A and transferred to bowl B. ∪ An ) = n+1 1 1 1 1 − 2! + 3! − 4! + . An urn contains 15 red balls and 15 white balls. if it is known that they are of the same color. Two socks are selected at random from the drawer. 24 35 58 + 58 11. . (d) The probability of at least 4! 1 1 1 one match is P(A1 ∪ A2 ∪ A3 ∪ A4 ) = 1 − 2! + 3! − 4! . Let the event Ai denote a match on the ith draw. 1 − ( 3 )4 4 (b) Without replacement. .draw a blue chip. . + (4) 1 (4) 1 2 + 4 (5) 3 (5) 1 2 4 16. (a) Compute the probability that both socks are of the same color. The ﬁrst person to select WIN ball is the winner. An urn contains ﬁve balls. The balls are selected one at a time. (e) Extend this exercise so that there are n balls in the urn. Compute the probability of (a) HTHTHHT.

scribd
/*********** DO NOT ALTER ANYTHING BELOW THIS LINE ! ************/ var s_code=s.t();if(s_code)document.write(s_code)//-->
user
The text passage is a sample quiz containing several probability and combinatorics problems, each with a question and an answer provided.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS