Solutions: Probability

Introduction to Probability

Exercise Problem Solution
1 Which of the following is an experiment? 

Tossing a coin.
Rolling a single 6-sided die.
Choosing a marble from a jar.
All of the above.

All of the above.
2 Which of the following is an outcome? 

Rolling a pair of dice.
Landing on red.
Choosing 2 marbles from a jar.
None of the above.

Landing on red.
3 Which of the following experiments does NOT have equally likely outcomes? 

Choose a number at random from 1 to 7.
Toss a coin.
Choose a letter at random from the word SCHOOL.
None of the above.

Choose a letter at random from the word SCHOOL (SCHOOL has 2 O's).
4 What is the probability of choosing a vowel from the alphabet? 

21_over_26.gif

5_over_26.gif

1_over_21.gif

None of the above.

5_over_26.gif
5 A number from 1 to 11 is chosen at random. What is the probability of choosing an odd number? 

1_over_11.gif

5_over_11.gif

6_over_11.gif

None of the above.

6_over_11.gif

 

Certain and Impossible Events

Exercise Problem Solution
1 A glass jar contains 5 red, 3 blue and 2 green jelly beans. If a jelly bean is chosen at random from the jar, then which of the following is an impossible event? 

Choosing a red jelly bean.
Choosing a blue jelly bean.
Choosing a yellow jelly bean.
None of the above.

Choosing a yellow jelly bean.
2 A spinner has 7 equal sectors numbered 1 to 7. If you spin the spinner, then which of the following is a certain event? 

Landing on a number less than 7.
Landing on a number less than 8.
Landing on a number greater than 1.
None of the above.

Landing on a number less than 8
3 What is the probability of choosing 14 hearts from a standard deck of 52 playing cards? 

14_over_52.gif

1

0

None of the above.

0; Each suit has only 13 cards.
4 If a number is chosen at random from the following list, then what is the probability that it is prime?
2, 3, 5, 7, 11, 13, 17, 19 

1

0

1_over_8.gif

None of the above.

1
5 If a single 6-sided die is rolled, then which of the following events is neither certain nor impossible? 

Rolling a number less than 7.
Rolling an even number.
Rolling a zero.
None of the above.

Rolling an even number.

 

Sample Spaces

Exercise Problem Solution
1 What is the sample space for choosing an odd number from 1 to 11 at random? 

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
{1, 3, 5, 7, 9 11}
None of the above.

{1, 3, 5, 7, 9 11}
2 What is the sample space for choosing a prime number less than 15 at random? 

{2, 3, 5, 7, 11, 13, 15}
{2, 3, 5, 7, 11, 13}
{2, 3, 5, 7, 9, 11, 13}
All of the above.

{2, 3, 5, 7, 11, 13}
3 What is the sample space for choosing 1 jelly bean at random from a jar containing 5 red, 7 blue and 2 green jelly beans? 

{5, 7, 2}
{5 red, 7 blue, 2 green}
{red, blue, green}
None of the above.

{red, blue, green}
4 What is the sample space for choosing 1 letter at random from 5 vowels? 

{a, e, i, o, u}
{v, o, w, e, l}
{1, 2, 3, 4, 5}
None of the above.

{a, e, i, o, u}
5 What is the sample space for choosing 1 letter at random from the word DIVIDE? 

{d, i, v, i, d, e}
{1, 2, 3, 4, 5, 6}
{d, i, v, e}
None of the above.

{d, i, v, e}

 

The Complement of an Event

Exercise Problem Solution
1 A glass jar contains 5 red, 3 blue and 2 green jelly beans. If a jelly bean is chosen at random from the jar, what is the probability that it is not blue? 

1_over_2.gif

3_over_10.gif

7_over_10.gif

None of the above.

1 - 3_over_10.gif = 7_over_10.gif

Answer: 7_over_10.gif

2 A student is chosen at random from a class of 16 girls and 14 boys. What is the probability that the student chosen is not a girl? 

8_over_15.gif

7_over_15.gif

1

None of the above.

1 - 16_over_30.gif = 14_over_30.gif = 7_over_15.gif

Answer: 7_over_15.gif

3 A number from 1 to 5 is chosen at random. What is the probability that the number chosen is not odd? 

2_over_5.gif

3_over_5.gif

0

None of the above.

1 - 3_over_5.gif = 2_over_5.gif

Answer: 2_over_5.gif

4 If a number is chosen at random from the following list, what is the probability that it is not prime?
2, 3, 5, 7, 11, 13, 17, 19 

1

1_over_8.gif

0

None of the above.

1 - 8_over_8.gif = 0

Answer: 0 (this is an impossible event)

5 If a single 6-sided die is rolled, what is the probability of rolling a number that is not 8? 

5_over_6.gif

1

0

None of the above.

P(8) = 0; P(not 8) = 1 - 0 = 1

Answer: 1 (This is a certain event)

 

Mutually Exclusive Events

Exercise Problem Solution
1 Which of the following are mutually exclusive events when a single card is chosen at random from a standard deck of 52 playing cards? 

Choosing a 7 or Choosing a club.
Choosing a 7 or Choosing a jack.
Choosing a 7 or Choosing a heart.
None of the above.

Choosing a 7 or Choosing a jack
2 All of the following are mutually exclusive events when a single 6-sided die is rolled EXCEPT: 

Rolling a number less than 4 or Rolling a number greater than 4.
Rolling a 2 or Rolling an odd number.
Rolling a 2 or Rolling an even number.
None of the above.

Rolling a 2 or Rolling an even number.
3 Which of the following are mutually exclusive events when a day of the week is chosen at random? 

Choosing a Monday or Choosing a Wednesday.
Choosing a Saturday or Choosing a Sunday.
Choosing a weekday or Choosing a weekend day.
All of the above.

All of the above.
4 A single letter is chosen at random from the word TEACHER. All of the following are mutually exclusive events except:

Choosing a T or Choosing a consonant.
Choosing a T or Choosing a vowel.
Choosing an E or Choosing a C.
None of the above.

Choosing a T or Choosing a consonant.
5 Which of the following are mutually exclusive events when a month of the year is chosen at random? 

Choosing August or Choosing a summer month.
Choosing September or Choosing a fall month.
Choosing a summer month or Choosing a winter month.
None of the above.

Choosing a summer month or Choosing a winter month.

 

Addition Rules for Probability

Exercise Problem Solution
1 A day of the week is chosen at random. What is the probability of choosing a Monday or Tuesday? 

1_over_7.gif

2_over_14.gif

2_over_7.gif

None of the above.

1_over_7.gif + 1_over_7.gif = 2_over_7.gif

Answer: 2_over_7.gif
 

2 In a pet store, there are 6 puppies, 9 kittens, 4 gerbils, and 7 parakeets. If a pet is chosen at random, what is the probability of getting a puppy or a parakeet? 

1_and_15_over_26.gif

1_over_2.gif

11_over_26.gif

None of the above.

6_over_26.gif + 7_over_26.gif = 13_over_26.gif = 1_over_2.gif

Answer: 1_over_2.gif

3 The probability of a New York teenager owning a skateboard is 0.37, of owning a bicycle is 0.81, and of owning both is 0.36. If a New York teenager is chosen at random, what is the probability that the teenager owns a skateboard or a bicycle? 

1.18
0.7
0.82
None of the above.

(0.37 + 0.81) - 0.36 = 0.82

Answer: 0.82

4 A number from 1 to 10 is chosen at random. What is the probability of choosing a 5 or an even number?

3_over_5.gif

1_over_2.gif

1_over_5.gif

All of the above.

P(5) = 1_over_10.gif

P(even) = 5_over_10.gif

P(5 or even) = 1_over_10.gif + 5_over_10.gif= 6_over_10.gif = 3_over_5.gif

Answer: 3_over_5.gif

5 A single 6-sided die is rolled. What is the probability of rolling a number greater than 3 or an even number?

1

2_over_3.gif

5_over_6.gif

None of the above.

P(greater than 3) = 3_over_6.gif

P(even) = 3_over_6.gif

P(both) = 2_over_6.gif

P(greater than 3 AND even)

= (3_over_6.gif + 3_over_6.gif) - 2_over_6.gif = 4_over_6.gif = 2_over_3.gif

Answer: 2_over_3.gif

 

Probability of Independent Events

Exercise Problem Solution
1 Spin a spinner numbered 1 to 7, and toss a coin. What is the probability of getting an odd number on the spinner and a tail on the coin? 

3_over_14.gif

2_over_7.gif

5_over_14.gif

None of the above.

4_over_7.gif · 1_over_2.gif = 4_over_14.gif = 2_over_7.gif

Answer: 2_over_7.gif

2 A jar contains 6 red balls, 3 green balls, 5 white balls and 7 yellow balls. Two balls are chosen from the jar, with replacement. What is the probability that both balls are green? 

6_over_441.gif

2_over_49.gif

1_over_49.gif

None of the above.

3_over_21.gif · 3_over_21.gif = 9_over_441.gif = 1_over_49.gif

Answer: 1_over_49.gif

3 In Exercise 2, what is the probability of getting a red and a yellow ball?

2_over_21.gif

3_over_21.gif

13_over_63.gif

All of the above.

6_over_21.gif · 7_over_21.gif = 42_over_441.gif

= 2_over_21.gif

Answer: 2_over_21.gif

4 Four cards are chosen from a standard deck of 52 playing cards with replacement. What is the probability of choosing 4 hearts in a row?

13_over_52.gif

1_over_16.gif

1_over_256.gif

None of the above.

13_over_52_0.gif · 13_over_52_0.gif · 13_over_52_0.gif · 13_over_52_0.gif

1_over_4.gif · 1_over_4.gif · 1_over_4.gif · 1_over_4.gif

= 1_over_256.gif

Answer: 1_over_256.gif

5 A nationwide survey showed that 65% of all children in the United States dislike eating vegetables. If 4 children are chosen at random, what is the probability that all 4 dislike eating vegetables? (Round your answer to the nearest percent.) 

18%
260%
2%
None of the above.

65% = .65;

(.65)(.65)(.65)(.65)

= .179

.179 rounded to the nearest percent is 18%

Answer: 18%

 

Probability of Dependent Events

Exercise Problem Solution
1 Two cards are chosen at random from a deck of 52 cards without replacement. What is the probability of getting two kings?

4_over_663.gif

1_over_221.gif

1_over_69.gif

None of the above.

4_over_52a.gif · 3_over_51.gif = 12_over_2652.gif

= 1_over_221.gif

Answer: 1_over_221.gif

2 Two cards are chosen at random from a deck of 52 cards without replacement. What is the probability that the first card is a jack and the second card is a ten?

3_over_676.gif

1_over_169.gif

4_over_663_1.gif

None of the above.

4_over_52a.gif · 4_over_51.gif = 16_over_2652.gif

= 4_over_663_1.gif

Answer: 4_over_663_1.gif

3 On a math test, 5 out of 20 students got an A. If three students are chosen at random without replacement, what is the probability that all three got an A on the test? 

1_over_114.gif

25_over_1368.gif

3_over_400.gif

None of the above.

5_over_20.gif · 4_over_19.gif · 3_over_18.gif =

60_over_6840.gif = 1_over_114.gif

Answer: 1_over_114.gif

4 Three cards are chosen at random from a deck of 52 cards without replacement. What is the probability of getting an ace, a king and a queen in order?

1_over_2197.gif

8_over_5525.gif

8_over_16575.gif

None of the above.

P(ace, king, queen)

= 4_over_52a.gif · 4_over_51.gif · 4_over_50.gif

64_over_132600.gif = 8_over_16575_1.gif

Answer: 8_over_16575_1.gif

5 A school survey found that 7 out of 30 students walk to school. If four students are selected at random without replacement, what is the probability that all four walk to school? 

343_over_93960.gif

1_over_783.gif

7_over_6750.gif

None of the above.

7_over_30.gif · 6_over_29.gif · 5_over_28.gif · 4_over_27.gif =

840_over_657720.gif = 1_over_783.gif

Answer: 1_over_783.gif

 

Conditional Probability

Exercise Problem Solution
1 In New York State, 48% of all teenagers own a skateboard and 39% of all teenagers own a skateboard and rollerblades. What is the probability that a teenager owns rollerblades given that the teenager owns a skateboard?

87%
81%
123%
None of the above.

point39_over_point48.gif = .8125

Answer: 81% (to the nearest percent)

2 At a middle school, 18% of all students play football and basketball, and 32% of all students play football. What is the probability that a student plays basketball given that the student plays football?

56%
178%
50%
None of the above.

point18_over_point32.gif = .5625

Answer: 56% (to the nearest percent)

3 In the United States, 56% of all children get an allowance and 41% of all children get an allowance and do household chores. What is the probability that a child does household chores given that the child gets an allowance? 

137%
97%
73%
None of the above.

point41_over_point56.gif = .7321

Answer: 73% (to the nearest percent)

4 In Europe, 88% of all households have a television. If it is known that 51% of all households have a VCR given that they also have a television, then what is the probability that a household has a VCR and a television?

58%
45%
173%
None of the above.

Let A denote the event "have a television" and let B denote the event "have a VCR".

conditional_soln_exercise4_step1.gif

conditional_soln_exercise4_step2.gif

conditional_soln_exercise4_step3.gif

Answer: 45% (to the nearest percent)

5 In New England, 84% of the houses have a garage and 65% of the houses have a garage and a back yard. What is the probability that a house has a backyard given that it has a garage? 

77%
109%
19%
None of the above.

point65_over_point84.gif = .7738

Answer: 77% (to the nearest percent)

 

Practice Exercises

Exercise Problem Solution
1 Which of the following is an impossible event?

Choosing an odd number from 1 to 10.
Getting an even number after rolling a single 6-sided die.
Choosing a white marble from a jar of 25 green marbles.
None of the above.

Choosing a white marble from a jar of 25 green marbles.
2 Which of the following is the sample space for choosing a letter from the word LIBRARY?

{I, A}
{L, I, B, R, A, R, Y}
{L, I, B, R, A, Y}
None of the above.

{L, I, B, R, A, Y}
3 What is the probability that a single card chosen from a deck is not an ace? 

4_over_52.gif

48_over_52.gif

39_over_52.gif

None of the above.

P(not ace) = 1 - P(ace)

= 1 - 4_over_52a.gif

= 48_over_52_0.gif

Answer: 48_over_52_0.gif

4 Which of the following is a certain event?

Choosing a teacher from a room full of students.
Choosing an odd number from the numbers 1 to 10.
Getting a 4 after rolling a single 6-sided die.
None of the above.

The probability of a certain event is equal to 1.

Answer: None of the above.

5 There are 4 parents, 3 students and 6 teachers in a room. If a person is selected at random, what is the probability that it is a teacher or a student? 

9_over_13.gif

4_over_13.gif

7_over_13.gif

None of the above.

P(teacher or student) = P(teacher) + P(student)

= 6_over_13.gif + 3_over_13.gif

= 9_over_13_0.gif

Answer: 9_over_13_0.gif
 

6 In a high school computer class there are 15 juniors and 10 seniors. Four juniors and five seniors are boys. If a student is selected at random, then what is the probability of selecting a junior or a boy? 

24_over_25.gif

four_fifths.gif

1_over_5.gif

None of the above.

P(junior or boy)

= P(junior) + P(boy) - P(junior and boy)

= 15_over_25.gif + 9_over_25.gif - 4_over_25.gif

= 20_over_25.gif = four_fifths.gif

Answer: four_fifths.gif

7 A jar contains 5 red, 3 green, 2 purple and 4 yellow marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. What is the probability of choosing a purple and then a red marble? 

5_over_98_1.gif

1_over_2_1.gif

3_over_98.gif

2_over_49_1.gif

P(purple, red) = P(purple) · P(red)

= 2_over_14_1.gif · 5_over_14_1.gif

= 10_over_196.gif

= 5_over_98_1.gif

Answer: 5_over_98_1.gif

8 Three cards are chosen at random from a deck without replacement. What is the probability of choosing an eight, a seven and a six, in order? 

6_over_35152.gif

1_over_2197_1.gif

8_over_16575_2.gif

None of the above.

P(eight, seven, six)

= 4_over_52a.gif · 4_over_51.gif · 4_over_50.gif

= 64_over_132600.gif

= 8_over_16575_2.gif

Answer: 8_over_16575_2.gif

9 In a shipment of 25 DVD Players, 2 are defective. If 2 DVD Players are randomly selected and tested, what is the probability that both are defective if the first one is not replaced after it has been tested? 

4_over_625.gif

1_over_300.gif

2_over_625.gif

None of the above.

P(defective, defective)

= 2_over_25.gif · 1_over_24.gif

= 2_over_600.gif

= 1_over_300.gif

Answer: 1_over_300.gif

10 In a school, 48% of the students take a foreign language class and 19% of students take both foreign language and technology. What is the probability that a student takes technology given that the students takes foreign language? (Round your answer to the nearest percent.) 

67%
253%
40%
None of the above.

Let FL represent foreign language

foreign_language.gif

= point19_over_point48.gif

= 0.3958333... 

= 40% rounded to the nearest percent

Answer: 40%

 

Challenge Exercises

Exercise Problem Solution
1 Which of the following is the sample space when 2 coins are tossed?

{H, T, H, T}
{H, T}
{HH, HT, TH, TT}
None of the above.

{HH, HT, TH, TT}
2 At Kennedy Middle School, 3 out of 5 students make honor roll. What is the probability that a student does not make honor roll? 

65%
40%
60%
None of the above.

3_over_5_1.gif = .60; 1 - .60 = .40

Answer: 40%

3 A large basket of fruit contains 3 oranges, 2 apples and 5 bananas. If a piece of fruit is chosen at random, what is the probability of getting an orange or a banana? 

4_over_5_1.gif

1_over_2_1.gif

7_over_10_1.gif

None of the above.

P(orange or banana)

= 3_over_10_1.gif + 5_over_10_1.gif

= 8_over_10.gif = 4_over_5_1.gif

Answer: 4_over_5_1.gif

4 A pair of dice is rolled. What is the probability of getting a sum of 2?

1_over_6.gif

1_over_3.gif

1_over_36.gif

None of the above.

P(1 and 1) = 1_over_6.gif · 1_over_6.gif

= 1_over_36.gif

Answer: 1_over_36.gif

5 In a class of 30 students, there are 17 girls and 13 boys. Five are A students, and three of these students are girls. If a student is chosen at random, what is the probability of choosing a girl or an A student? 

19_over_30.gif

11_over_15.gif

17_over_180.gif

None of the above.

P(girl) + P(A) - P(girls and A)

= 17_over_30.gif + 5_over_30.gif - 3_over_30.gif

= 19_over_30.gif

Answer: 19_over_30.gif

6 In the United States, 43% of people wear a seat belt while driving. If two people are chosen at random, what is the probability that both of them wear a seat belt? 

86%
18%
57%
None of the above.

(.43)(.43) = .1849

Answer: 18% (to the nearest percent)

7 Three cards are chosen at random from a deck without replacement. What is the probability of getting a jack, a ten and a nine in order? 

8_over_16575_2.gif

1_over_2197_1.gif

6_over_35152.gif

None of the above.

P(jack, ten, nine)

solns_challenge7_step1.gif

8_over_16575_2.gif

Answer: 8_over_16575_2.gif

8 A city survey found that 25% of teenagers have a part time job. The same survey found that 40% plan to attend college. If a teenager is chosen at random, what is the probability that the teenager has a part time job and plans to attend college? 

65%
15%
10%
None of the above.

(.25)(.40) = .10

Answer: 10%

9 In a school, 14% of students take drama and computer classes, and 67% take drama class. What is the probability that a student takes computer class given that the student takes drama class? 

81%
21%
53%
None of the above.

 = .2090

Answer: 21% (to the nearest percent)

10 In a shipment of 100 televisions, 6 are defective. If a person buys two televisions from that shipment, what is the probability that both are defective? 

3_over_100.gif

9_over_2500.gif

1_over_330.gif

None of the above.

P(defective, defective)

= 6_over_100.gif · 5_over_99.gif

= 30_over_9900.gif

= 1_over_330.gif

Answer: 1_over_330.gif