Combinatorics and probabilities MATHEMATICS

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SALES AND MARKETING Department

MATHEMATICS

2 ^nd Semester

Combinatorics and probabilities ^

Tutorials and exercises

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Exercise 1. (Tutorial for lesson page 8)

E is the set of the inhabitants of a city ; Card(E) = 2500.

A is the set of this city’s men ; Card(A) = 1220.

B is the set of this city’s retired people ; Card(B) = 670.

400 women are retired.

Create, then complete, a contingency table; tell how many men aren’t retired; tell how many people are women or retired people.

Exercise 2.

Given A and B, two subsets of E, simplify the following expressions:

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A∩ ∪ ∩A B B ^A^{∩ ∪ ∩}

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^A ^B

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^B

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^A^{∪ ∩}^B

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^A^∩^B

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^A^{∩ ∪}^B

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^A^∪^B

Exercise 3.

E = {2, 5, 8, 11, 14, 17, 20}. Let A be the subset of the even numbers of E and B the one of the multiples of 5.

1) Define the complement of A in E. Give its elements.

2) Give the sets A∩B and A∩B. What is their union?

Exercise 4.

There are three categories of water meters in a municipality:

A: Meters that are less than two years old and are therefore still under warranty;

B: Meters that are between 2 and 20 years old;

C: Meters that are more than 20 years old.

Let D be the set of defective water meters.

1) Which of these sets are mutually exclusive?

2) What do the following sets represent?

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A ; D ; A∩D ; A∪D ; C∩D ; A∪D ; A∩ ∪D A∩D ; A∪B Exercise 5.

In a group of 25 students (including 17 women), 20 passed their exam (including 14 women).

1) Build a contingency table dispatching the information above.

2) How many men passed their exam? What is the name of the corresponding set?

Exercise 6.

Amongst 350 people interviewed for a survey, 244 own a computer with an Internet access, 287 own a smartphone, but inside this last group, 56 don’t have a computer with an Internet access.

1) Organize and fill a contingency table using these data.

2) How many people own…

a. an Internet access but no smartphone ? b. at least one of both?

c. only one of both?

Exercise 7.

After counting the answers from a survey conducted on a sample of 500 people, it appears that 154 of them go to the cinema at least once a month, 228 buy popcorn when they go to movies, and of those who go to the cinema less than once a month, 131 usually buy popcorn.

1) Let’s name A the set of people who go to the cinema at least once a month and B the set of people who buy popcorn when they go to movies; build the corresponding contingency table.

2) Answer by naming the adequate subset and justifying the result if not already written in the table:

a. Among people who go to the cinema at least once a month, how many buy popcorn ? b. How many people go to the cinema less than once a month?

c. How many people are elements of A or B ?

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IUT de Saint-Etienne – Département TC –J.F.Ferraris – Math – S2 – CombProb – TEx – Rev2020 – page 2 / 9

* by how many ways can we arrange two objects inside three drawers?

* how many numbers composed with four figures only contain the figures 1, 2, 3?

* how many words can be written by taking five letters chosen in the set {a, b, e, m, i, r, o}?

* how many pairs representative/assistant could have been elected from a group of 25 students?

* how many ways can 3 blocks be piled, taking them among 10 blocks of different colours?

* how many words can be written by taking five different letters chosen in {a, b, e, m, i, r, o}?

* How many couples of representatives could be elected from a group of 25 students?

* How many different hands of 8 cards could be given from a deck of 32 playing cards?

* How many draws of 6 different integers are possible, taking them between 1 and 49?

1) From a deck of 32 playing cards, how many 8-card hands own exactly 3 spades and 2 hearts?

2) In a company, among 20 women and 20 men, 5 women and 3 men have to be chosen at random. How many possibilities are there?

Exercise 12.

1. dice

1.1 A die is being rolled three times. How many possible outcomes?

1.2 Three dice are being rolled at the same time. How many possible outcomes?

2. numbers and letters

2.1 How many phone numbers of eight figures can theoretically exist ? 2.2 How many ways can six different integers be chosen among [1 ; 49]?

2.3 How many numbers are composed with three different figures (including 0)?

2.4 How many different lists of 4 letters can be created for the vehicles’ number plates?

2.5 How many anagrams of the word "MATHS" are there?

2.6 How many words can be created, taking 4 letters from the word "BRACKET"?

2.7 How many 10 notes-long melodies can be written, taking notes among A,B,C,D,E,F,G?

3. arrangements

3.1 How many ways can 5 objects be arranged into 8 boxes?

3.2 Same question, but you can’t place more than one object per box.

3.3 Paul drives a team of 5 people. Each month, he evaluates the work of one of them, chosen at random.

In a one year period, how many different lists of evaluated people could have been made?

4. playing cards

4.1 How many 8 cards hands from a deck of 32 playing cards?

4.2 How many 5 cards hands from a deck of 52 playing cards?

5. classifications, elections

5.1 How many possible different tiercés at the end of a 12 horses race?

5.2 How many possible different classifications at the end of a 12 horses race?

5.3 How many ways a delegation of 5 people can be chosen from a group of 40?

5.4 16 pilots are fighting at a formula 1 race. At the end, the only six first will score different numbers of points. How many possible distributions of points are there?

5.5 How many possible podiums after a race in which 8 runners will compete?

5.6 Fifteen people meet. Everyone gives a handshake to every other, once. How many handshakes?

Exercise 13.

Let consider a group of 13 women and 8 men. 4 people have to be chosen.

1) How many possibilities are there?

2) How many of them contain exactly one man?

3) 2 men? 3 men? 4 men? no man?

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Exercise 14.

Among the hands of 5 cards taken from a deck of 32 playing cards, how many contain:

a. the 4 aces? b. a square? c. exactly 3 spades?

d. exactly 2 spades and one club? e. at least one king?

f. at least two jacks? g. exactly 3 diamonds and one king?

Exercise 15.

A jar contains balls: 2 white, 3 green, 5 red. 3 balls are to be taken simultaneously from the jar. Among the possible groups of 3 balls, how many contain…

a. one single colour? b. the three colours? c. two colours? d. at least two colours?

Exercise 16.

On loto game (former rules), a player has to select 6 different numbers from the set {1, 2, 3, …, 49}. Then, the official draw is performed, revealing the 6 winner numbers.

1) Calculate the whole number of possible selections.

2) Among them, how many would contain exactly…

a. the six winner numbers? b. 5 winner numbers? c. 4 winner numbers?

d. 3 winner numbers? e. no winner number? (find two ways ) Exercise 17.

1) Independent questions

a. In order to build your team, you have to choose 5 people out of a group of 10 basket players. How many different teams could be made?

b. A company has 18 employees and its manager decides to give three awards : best employee, most punctual employee, and less bald employee. How many ways can these awards be given?

c. On a chess board (8 × 8 tiles), how many ways can you put a king, a queen and a tower ?

2) 20 chips have been put in a bag, numbered from 1 to 20. The chips from #1 to 10 are white; those from #11 to 16 are green; the last ones are red. You have to draw three chips, at random, simultaneously.

a. How many different possible draws are there?

b. How many draws can be made with three white chips?

c. How many draws would show three different colours?

d. How many draws would show three chips of the same colour ? e. How many draws would show three even chips of the same colour?

A random experiment consists in taking one integer, at random, among {1 ; … ; 20}.

1) Determine Card(Ω).

2) Let’s name some events: A : "get at least 15" and B : "get an even number". Determine : p(A) , p(A) , p(B) , p(A∩B) , p(A∪B).

A random experiment consists in a simultaneous drawing of 3 letters in our alphabet.

1) Determine Card(Ω)

2) b. We set the following events A: "get 3 consonants", B: " get 2 consonants", C: " get 1 consonant"

and D: "get 3 vowels"

a. Are they mutually exclusive?

b. Do they represent a partition of Ω?

c. Calculate their cardinal numbers and then their probabilities (writing four significant figures). Finally, check the sum of their cardinal numbers and the sum of their probabilities.

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Random experiment: roll two dice and add both results.

1) What are the different possible sums?

2) Are they equally likely?

3) Build a sample space of equally likely outcomes.

4) We set the following events: A: "the sum equals 10" and B:

"the sum is at least 10".

Determine p(A) ; p(A) ; p(B).

E = {1, 2, 3, …, 10}, A: "even numbers of E", B : "multiples of 3 in E"

Venn diagram: probabilistic choice tree:

E

2 4 6

8 10 A

1 5 3 B

7 9

contingency table:

A A Inside, cardinal numbers of the

corresponding intersections must be placed, subtotals ("marginal

frequencies") and the overall total

B = Card(B)

= Card(A) = Card(A) = Card(E) 1) Complete the contingency table.

2) Let’s choose a number between 1 and 10, at random, not looking at it.

a. What is the probability it would be even?

b. What is the probability it would be a multiple of 3 ?

3) Let’s choose a number between 1 and 10, at random, not looking at it, but we’re told it’s a multiple of 3.

a. What is the probability it would be even?

b. What is the result obtained by the corresponding formula?

4) Let’s choose a number between 1 and 10, at random, not looking at it, but we’re told it’s even.

a. What is the probability it would be a multiple of 3 ? b. What is the result obtained by the corresponding formula?

A laboratory has developed a breathalyzer. A reliability test has been done on this product, with a test- population on which it's been stated that 2% exceed 0.5 g/L (event E) and so are out of law. Everyone exhales into the breathalyzer; the event P refers to a positive result given by this device.

The reliability test has given the following results:

- 95% of people who really exceed 0.5 g/L got a positive result by the breathalyzer;

- 96% of people who don't exceed 0.5 g/L got a negative result by the breathalyzer.

What is then your probability of really exceeding 0.5 g/L, given that your result is positive?

1) Taking back exercise 22: are E and P independent?

2) Taking back exercise 18: are A and B independent?

Exercise 24.

3 dice are rolled together.

1) How many different outcomes are possible?

2) Calculate the probabilities of the following events:

a. "get a triple 6" b. "get a triple" c. "get a 421" d. "get at least one 4" e. "get a sum of 10"

3) a. Are B and D mutually exclusive? b. Are C and E mutually exclusive?

2;4;6;8;10

A

1;3;5;7;9

A

B

6

B

3;9 2;4;8;10

B

1;5;7

B

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Exercise 25.

18 balls lay in a jar: 7 white, 9 red, 2 green. Three balls are simultaneously taken out.

1) What is the probability to see the three colours in your hand?

2) What is the probability to see only one colour in your hand?

3) a. Which third event makes a partition of Ω with both former ones? b. Deduce its probability.

4) Calculate the probability to get no white ball or no green ball.

Exercise 26.

A bag contains 20 coins: n are black and the 20-n others are white. 2 coins have to be picked up together.

1) Express (with n) the probabilities of the following events:

a. A: One black and one white. b. B: two black c. C: two white 2) Check that the global probability equals 1.

3) Determine, by solving an equation, the values of n such that p(C) > 0.5.

Exercise 27.

A taxi is involved in a pile-up at night. Two taxi companies, the Green and the Blue, operate in the city. We have the following data:

(a) 85 % of the taxis (in the city) are Green and 15 % are Blue.

(b) A witness identified the taxi responsible as a Blue.

The court tested the reliability of the testimony in such circumstances (accident at night) and concluded that the witnesses correctly identified the colours in 80% of cases and were wrong in 20% of cases.

What is the probability that the taxi involved in the accident was a Blue?

Exercise 28.

A bank found that 2% of the checks issued by its clients aren't correctly worded (correctly worded : event W).

97% of correctly worded checks are correctly entered by the agent into the bank's data base (event E).

When it's not correctly worded, the agent is able to correct the mistakes 5 times out of 100.

A check has to be entered into data base. calculate the probabilities of the following events : 1) The agent doesn’t enter the check correctly.

2) The check has been correctly worded, given that the agent entered it correctly

3) The check has not been correctly worded, given that the agent didn’t enter it correctly Exercise 29.

35 % of people in a city are employees. Among them, 8 people on 10 use their car every day, whereas 30 % of the unemployed do (employee: event E; use car every day: event D). We are to select one person in this city, at random.

1) Display the different categories of people in a probabilistic choice tree.

2) a. What is the probability that this random individual be unemployed?

b. What is the probability that this person be an employee who uses his/her car every day?

c. What is the probability that he/she uses his/her car every day?

d. Given that this person drives his/her car every day, probability he/she is an employee?

e. Are the events E and D independent?

Exercise 30.

Following the discovery in one country of the first cases of a contagious disease, a major vaccination campaign was carried out: 70% of the inhabitants were vaccinated. A study revealed that 5% of the vaccinated had been affected to varying degrees by the disease, while 60% of the unvaccinated had been affected to varying degrees. Calculate:

1) The probability that an individual randomly selected from the population has been affected by this disease.

2) The probability that an individual has been vaccinated, given that he or she has been affected.

Exercise 31.

Four girls and three boys are required to take an oral examination. The examiner decides to draw up a random list of candidates establishing the order in which they are to take the test. To do this, he puts the names (all supposedly different) of the seven candidates in an envelope and then draws the seven names one after the other.

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Let’s name F1 the event: "The first candidate interviewed was a girl", and F2 the event: "The second candidate interviewed was a girl".

1) What is the probability that the first two candidates interviewed are girls?

2) What is the probability that the first candidate interviewed is a girl given that the second candidate interviewed is a girl?

Exercise 32.

During a TV game show, a contestant stands in front of three boxes, only one of which contains the jackpot, and chooses one box at random. Before opening it, the presenter points out one of the other two boxes that does not contain the jackpot.

The presenter then suggests two strategies to the candidate:

STRATEGIE 1 : the candidate maintains his or her first choice;

STRATEGIE 2 : the candidate does not maintain his or her initial choice and chooses the door not designated by the presenter.

Which one is the best strategy for the player?

A lottery is held. 100 tickets are to be sold, €1 each. One ticket is a €30 winner, two are €15 winners, and seven would make the buyer win €1. Considering we want to purchase one ticket, X is the random variable of the net gain (the €1 expense taken into account).

1) Give the probability distribution of X.

2) If we’re playing this lottery the same way many times, can we expect to be a long-term winner?

(begin by an estimate of what would be likely to occur after a thousand attempts) Exercise 34. (Tutorial for lesson page 16)

1) Calculate the expected value and the standard deviation, with the data of exercise 33. Comment.

2) If the possible gains and loss were doubled (in the initial array), what would these parameters become?

3) If the values of X were increased by € 0.5, what would these parameters become?

Exercise 35.

There are two hospitals in one city. In the larger one, about 45 babies are born every day, while in the smaller one, about 15 babies are born every day. As you know, about 50% of all babies are boys. However, the exact percentage on any given day varies: sometimes it is more than 50%, sometimes less.

Over a period of one year, each hospital recorded the days on which more than 60% of babies born were boys.

In your opinion, which hospital recorded the most days of this type?

Exercise 36.

A game consists of two identical boxes each having 10 chips numbered 1 through 10. The experiment is to pick a chip in each box.

1) a. Describe one of the possible outcomes.

b. Explain why the sample space’s cardinal number is 100.

c. What is the probability of choosing two even numbers?

d. Prove that the probability of two different even numbers is 0.2.

2) For one game, you have to spend € 1. If you get two different even numbers, you win € 1; if you get two identical numbers except 1 and 1, you earn € 6; 1 and if you get the double one, you win € 50; in all other cases, no gain. The random variable X gives the gain at the end of the game, regardless of the initial € 1 bet.

a. Give the probability distribution of X. b. Give the expected value of X. c. Can we expect to win money on playing this game a lot?

Exercise 37.

A bag contains 5 white and 10 black balls. You bet € 2 for a 3 balls draw together. Get 3 white makes you earn

€ 100; 2 white: € 10; 1 white: € 2; 3 black: nothing. The random variable X is your gain at the end of a test, once deduced the bet.

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2) Give the expected value and the standard deviation of X. 3) If you play a hundred times, what gain is the most likely?

Exercise 38.

A game consists of a random draw of a letter from the alphabet (which contains 20 consonants and 6 vowels A, E, I, O, U, Y). Each letter is assigned a number according to the following table:

A B C D E F G H I J K L M

1 2 3 4 5 6 7 8 9 10 11 12 13

N O P Q R S T U V W X Y Z

14 15 16 17 18 19 20 21 22 23 24 25 26

We'll set the events C: "the letter is a consonant" and M: "its number is at least 17".

Part 1

1) Build a probabilistic choice tree (1st level: C and its contrary; 2nd level: M and its contrary) into which the simple, conditional and intersection probabilities will be placed.

2) Given that a vowel has been drawn, what’s the probability its number is more than 16?

3) Given that its number is more than 16, what’s the probability it’s a vowel?

4) Are the events M and C independent?

Part 2

The event C∩M, is awarded a € 10 gain and the event C ∩M would make you lose € 5; as for the other possibilities: they don’t lead to either gain or loss. X is the random variable “gain after one draw”.

2) Give the expected value of X and its meaning.

3) Give the standard deviation of X and its meaning.

From a jar that contains 7 white balls and 3 black balls, let’s draw two balls, one after the other and without putting the first one back. We name X₁ the random variable corresponding to 1 point in case the first ball is black and 0 point in case it’s white ; we name X₂ the random variable corresponding to 1 point in case the first ball is black and 0 point in case it’s white.

a. Give p(X₁ = 0) and p(X₁ = 1).

b. Give pX1 = 0(X2 = 0) and pX1 = 0(X2 = 1), then pX1 = 1(X2 = 0) and pX1 = 1(X2 = 1).

c. Complete the probabilistic choice tree and then the associated probability table.

X₁

X₂ PX2

PX1 1

d. Is the knowledge of both marginal distributions sufficient for the knowledge of the joint distribution?

e. Compare p(X₁ = 0)×p(X₂ = 0) to p((X₁ = 0)∩(X₂ = 0)). Are the variables X₁ and X₂ independent?

Exercise 40.

Two sales agents A and B of a cooperative work in team for two weeks to obtain orders from potential customers. A is responsible for placing new contracts to existing customers and B is responsible for prospecting new customers.

Let’s name: X_A the random variable measuring the number of contracts obtained by A and X_B the random variable measuring the number of contracts obtained by B.

It’s assumed that X_A can only take its values in {0 ; 1 ; 2 ; 3} and X_B in {0 ; 1}.

The joint distribution of X_A and X_B is given through the following table:

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IUT de Saint-Etienne – Département TC –J.F.Ferraris – Math – S2 – CombProb – TEx – Rev2020 – page 8 / 9 X_A

X_B

0 1 2 3

0 0.05 0.15 0.20 0.10

1 0.1 0.2 0.15 0.05

1) a. Determine the margin distributions of X_A and of X_B. b. Are these variables independent?

2) Let’s set a new variable, X “total number of obtained contracts”, by X = X_A + X_B. a. Give the probability distribution of X.

b. Calculate E(X) and V(X).

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Combinatorics and probabilities MATHEMATICS

SALES AND MARKETING Department

MATHEMATICS

2 nd Semester

________ Combinatorics and probabilities ________

Tutorials and exercises

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IUT TC MATHEMATICS FORM "COMBINATORICS AND PROBABILITIES"

2 ^nd Semester

Combinatorics and probabilities ^

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