Exercise 1

In a class in which all students practice at least one sport, 60% of students play soccer or basketball and 10% practice both sports. If there is also 60% that do not play soccer, calculate the probability that a student chosen at random from the class:

1  Plays soccer only.

2 Plays basketball only.

3 Plays only one of the sports.

4 Plays neither soccer nor basketball.

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Exercise 2

In a city, 40% of the population have brown hair, 25% have brown eyes and 15% have both brown hair and eyes. A person is chosen at random.

1 If they have brown hair, what is the probability that they also have brown eyes?

2 If they have brown eyes, what is the probability of them not having brown hair?

3 What is the probability of them having neither brown hair nor brown eyes?

Exercise 3

There are two boxes. Box A contains 6 red balls and 4 blue balls and Box B contains 4 red balls and 8 blue balls. A die is rolled, if the number is less than 3, a ball is selected from box A. If the result is 3 or more, a ball is selected from Box B. Calculate:

1 The probability that the ball will be red and selected from Box B.

2 The probability that the ball will be blue.

Exercise 4

In order to write an exam, a student needs an alarm clock to wake up, which has proven to successfully wake him 80% of the time. If he hears the alarm in the morning the probability of writing the test is 0.9 and if he doesn´t hear the alarm the probability is 0.5.

1  If he writes the test what is the probability that he heard the alarm clock?

2 If he doesn´t write the test: what is the probability that he didn´t hear the alarm clock?

Exercise 5

On a shelf there are 60 novels and 20 poetry books. Person A chooses a book at random off the shelf and leaves with it. Shortly after, Person B chooses another book at random. Calculate:
1  The probability that the book selected by Person B is a novel?

2 If it is known that Person B chose a novel: what is the probability that the book selected by Person A was a poetry book?

Exercise 6

It is determined that 25 of every 100 men and 600 of every 1,000 women wear glasses. If the number of women in a particular room is four times more than that of men, calculate the probability of:

1 A person without glasses being randomly selected.

2 A woman with glasses being randomly selected.

Exercise 7

There are three sets of key rings A, B and C, for a house. The first set has five keys, the second has seven and the third has eight, of which only one key in each set opens the door to the store room. A keychain is chose at random followed by a key from the set.

1 What is the probability of the selected key being able to open the store room?

2 What is the probability that the chosen key chain is from the third set and the key does not open the door?

3 What is the probability that the chosen key opens the door and it came from the first key chain?

 

Solution of exercise 1

In a class in which all students practice at least one sport, 60% of students play soccer or basketball and 10% practice both sports. If there are  60% students who do not play soccer, calculate the probability that a student chosen at random from the class:

1  Plays soccer only.

Venn diagram - Exercise 2

P(S) = 1 - 0.6 = 0.4

P (S - \overline {B}) = 0.4 - 0.1 = 0.3

2 Plays basketball only.

P (B - \overline{S}) = 0.3 - 0.1 = 0.2

3 Plays only one of the sports

P(S - \overline {B}) U P (B - \overline{S}) = 0.3 + 0.2 = 0.5

 

4  Plays neither soccer nor basketball

P(\overline{F} - \overline{B}) = P (\overline {F U B}) = 1 - P (F U B) = 1 - 0.6 = 0.4

 

Solution of exercise 2

In a city, 40% of the population have brown hair, 25% have brown eyes and 15% have both brown hair and eyes. A person is chosen at random.

1  If they have brown hair, what is the probability that they also have brown eyes?

p (Brown eyes | Brown hair) = \frac{15}{40} = 0.375

 

2  If they have brown eyes, what is the probability of them not having brown hair?

p (No brown hair | Brown eyes) = \frac{10}{25} = 0.4

 

3  What is the probability of them having neither brown hair nor brown eyes?

p (neither brown hair or eyes) = \frac{50}{100} = 0.5

 

Solution of exercise 3

There are two boxes. Box A contains 6 red balls and 4 blue balls and Box B contains 4 red balls and 8 blue balls. A die is rolled, if the number is less than 3, a ball is selected from box A. If the result is 3 or more, a ball is selected from Box B. Calculate:

1 The probability that the ball will be red and selected from Box B.

Probability tree - Exercise 5

p (R \cap U_B) = \frac{4}{6} \cdot \frac{4}{12} = \frac{2}{9}

 

2   The probability that the ball will be blue.

p (Blue ball) = \frac{2}{6} \cdot \frac{4}{10} + \frac{4}{6} \cdot \frac{8}{12} = \frac{26}{45}

 

Solution of exercise 4

In order to write an exam, a student needs an alarm clock to wake up, which has proven to successfully wake him 80% of the time. If he hears the alarm in the morning the probability of writing the test is 0.9 and if he doesn´t hear the alarm the probability is 0.5.

If he writes the test what is the probability that he heard the alarm clock?

Probability tree diagram - Exercise 6

p (Heard |Test) = \frac{0.8 \cdot 0.9}{0.8 \cdot 0.9 + 0.2 \cdot 0.5} = \frac{36}{41}

 

2  If he doesn´t write the test: what is the probability that he didn´t hear the alarm clock?

p (No heard|No test) = \frac{0.2 \cdot 0.5} {0.8 \cdot 0.1 + 0.2 \cdot 0.5} = \frac{5}{9}

 

Solution of exercise 5

On a shelf there are 60 novels and 20 poetry books. Person A chooses a book at random off the shelf and leaves with it. Shortly after, Person B chooses another book at random. Calculate:

 

1 The probability that the book selected by Person B is a novel?

Probability tree diagram - Exercise 7

p (B novel) = \frac{60}{80} \cdot \frac{59}{79} + \frac {20}{80} \cdot \frac{60}{79} = \frac{237}{316}

 

2  If it is known that Person B chose a novel: what is the probability that the book selected by Person A was a poetry book?

p (A poetry | B novel) = \frac {60}{237}

 

Solution of exercise 6

It is determined that 25 of every 100 men and 600 of every 1,000 women wear glasses. If the number of women in a particular room is four times more than that of men, calculate the probability of:

1  A person without glasses being randomly selected.

Probability tree diagram - Exercise 8

p (No glasses) = \frac{1}{5} \cdot \frac{75}{100} + \frac {4}{5} \cdot \frac{400}{1000} = 0.47

 

2  A woman with glasses being randomly selected.

p (Woman with glasses) = \frac{4}{5} \cdot \frac {600}{1000} = 0.48

 

Solution of exercise 7

There are three sets of key rings A, B and C, for a house. The first set has five keys, the second has seven and the third has eight, of which only one key in each set opens the door to the store room. A keychain is chose at random followed by a key from the set.

1  What is the probability of the selected key being able to open the store room?

Probability tree - Exercise 9

p (Open) = \frac{1}{3} \cdot \frac{1}{5} + \frac{1}{3} \cdot \frac{1}{7} + \frac{1}{3} \cdot \frac{1}{8} = 0.1559

 

2  What is the probability that the chosen key chain is from the third set and the key does not open the door?

p (C and not open) = \frac{1}{3} \cdot \frac{7}{8} = 0.2917

 

3  What is the probability that the chosen key opens the door and it came from the first key chain?

p (A|open) = \frac{\frac{1}{3} \cdot \frac{1}{5}} {\frac{1}{3} \cdot \frac{1}{5} + \frac{1}{3} + \frac{1}{7} + \frac{1}{3} \cdot \frac{1}{8}} = 0.4275

 

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.