semester, Test Probabilities length: 2 hours – 20 points, inner coefficient 1.5

2019-2021 – S2 – Mathematics – TEST Probabilities SOLUTIONS – page 1 / 3

IUT of Saint-Etienne – Sales and Marketing department

Mr Ferraris Prom 2019-2021 25/05/2020

MATHEMATICS – 2

semester, Test Probabilities length: 2 hours – 20 points, inner coefficient 1.5

SOLUTIONS

Exercise 1: MCQ (2 points) – tick your answers below

1 good answer for each question; wrong, multiple or missing answer: 0 point 1) Which one displays the correct order?

C_n^p ≤P_n^p ≤n^p n^p ≤P_n^p ≤C_n^p n^p ≤C_n^p ≤P_n^p C_n^p ≤n^p ≤P_n^p 2) Rolling several dice at the same time inevitably leads to:

permutations p-lists combinations it depends

3) If A and B each have a 50% chance of occurring, then the chances of A∩B are:

0% 25% 50% it depends

4) On a choice tree, which of the following probabilities can be read directly?

events and conditional conditional and the three

intersections and events intersections previous kinds

Exercise 2: (2 points) Sets and cardinal numbers

By adapting the formula ^{Card A}

(

)

( )

( )

(

)

( )

Card A∪ ∪B C , thus a formula using only simple sets and intersections.

For the convenience of writing on a computer, you will use lowercase n for the intersection and lowercase u for the union, for example: Card(AnB), Card(AuB).

( )

( ) ( ) ( ) ( ( ) )

( ) ( ) ( ) ( )

( )

( ) ( ( ) ( ) ) ( ) ( ) ( )

Card A B C Card A Card B C Card A B C

with Card B C Card B Card C Card B C

and Card A B C Card A B A C Card A B Card A C Card A B C

∪ ∪ = + ∪ − ∩ ∪

∪ = + − ∩

∩ ∪ = ∩ ∪ ∩ = ∩ + ∩ − ∩ ∩

Thus:

( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

Card A∪ ∪B C =Card A +Card B +Card C −Card B∩ −C Card A∩ −B Card A∩ +C Card A∩ ∩B C

Exercise 3: (6 points) Combinatorics

1) How many integers, between 10,000 and 99,999, contain neither 0, 1, nor 2? 0.75 pt

2520 7776 15625 16807

An outcome is a number of five digits. Creating an outcome means choosing five digits (p = 5) in a set of seven (from 3 to 9) (n = 7).

In an outcome, the repetition of a digit is possible and their order has to be taken into account.

Therefore, an outcome is a p-list. The number of outcomes is n^p = 7⁵ = 16,807.

2) How many integers, between 0 and 99,999, contain neither 0, 1, nor 2? 1 pt

3619 9330 19607 20515

Here, we have to cumulate the number of integers of 1 digit, of 2 digits, of 3 digits, of 4 digits and of 5 digits.

In this case: 7¹ + 7² + 7³ + 7⁴ + 7⁵ = 7×(7⁵ – 1)/6 = 19,607.

3) Consumers are asked to choose their four favourite products from a list of twelve products, listing the four in order of preference. How many different rankings are possible? 1 pt

2019-2021 – S2 – Mathematics – TEST Probabilities SOLUTIONS – page 2 / 3

An outcome is a list of four products. Creating an outcome means choosing four products (p = 3) in a set of twelve (n = 12).

In an outcome, the repetition of a product is impossible and their order has to be taken into account.

Therefore, an outcome is a permutation. The number of outcomes is P = 11,880. ₁₂⁴

4) There are 15 stores on one street, 5 of which are clothing stores and 3 of which are grocery stores. The town hall decides to award a bonus to three shops whose names will be drawn at random from among the 15.

a. How many different draws of three stores are possible? 0.75 pt

An outcome is a set of four stores. Creating an outcome means choosing three stores (p = 3) in a set of fifteen (n = 15).

In an outcome, the repetition of a store is impossible and their order is not important.

Therefore, an outcome is a combination. The number of outcomes is C = 455. ³₁₅

b. Of these possible draws, how many include two clothing stores and one grocery store? 1.25 pt The stores are divided into three categories: clothing stores, grocery stores and other.

Let's display the set S of the 15 stores and the desired draws D:

S D

The number of such possible draws D is:

2 1 0

5 3 7

C × ×C C =10 3 1× × =30. clothing

grocery

5 3

2 1

other 7 0

total 15 3

c. Which is more likely: that the future draw has 1 clothing store or 2 clothing stores? (your answer will of

course have to be justified by calculations) 1.25 pt

The above reasoning can be applied to several types of draws:

S D1 D2

clothing 5 1 2

other 10 2 1

total 15 3 3

number of draws 225 100

It’s more likely to draw 1 clothing store out of the 3 chosen, than 2.

Exercise 4: (2 points) Conditional probabilities

A random experiment consists of rolling a die and noting its result.

Give an example of two compatible (= not mutually exclusive) but independent events (and justify).

Let’s set A: “even number” and B: “at least 3”. A and B can be realised at the same time (with a 4 or a 6), so they are compatible. p(A) = 3/6 and pB(A) = 2/4 = p(A), so A and B are independent.

Exercise 5: (3 points) Conditional probabilities

10% of French people over the age of 18 are left-handed (event "L"). 70% of them have passed their baccalaureate with honours ("H" event), compared with 40% of right-handed people.

Let’s build a probabilistic choice tree using the above information, adding the probabilities of intersections of L, H and their contrary events.

0.7 H 0.07

0.1 L

0.3 H 0.03 0.4 H 0.36

0.9 L

0.6 H 0.54

1) A student has just seen his baccalaureate results and exclaims, "I got it with honours! ". What is the

probability that this student is left-handed? 2 pts

2019-2021 – S2 – Mathematics – TEST Probabilities SOLUTIONS – page 3 / 3 pH(L) =

( )

( ) ( )

( ) ( )

p L H p L H 0.07

0.1628 16 28%

p H p L H p L H 0.07 0.36

∩ ∩

= = ≈ ≈

∩ + ∩ + .

There's a 16.28% chance this student is left-handed.

2) Are the events L and H independent? 1 pt

( )

( ) ( )

p L =0 1 ; p L ≈0.1628≠p G . That means L and H are not independent.

or : ^{p L}

(

)

( ) ( )

(

)

Exercise 6: (5 points) Simple probabilities and distributions

Let's take again an example seen in tutorial: the roll of two dice at the end of which we note the total of the two dice. Remember that the 36 possible pairs {(1,1), (1,2), ..., (3,3), (3,4), (3,5), ..., (5,6), (6,6)} are equally likely to occur and that, for example, the probability of making a total of 9 is p(9) = 4/36, because 4 pairs (out of the 36) give a total of 9: (3,6), (4,5), (5,4), (6,3).

We note A the event "obtain a total strictly lower than 8", B the event "obtain a total of 8 or 9" and C the event "obtain a total strictly higher than 9".

1) a. Give the probabilities of the events A, B and C. 0.75 pt

The table opposite cross-references the possible values of each die and adds them. Thanks to this, we see that 21 pairs lead to event A, 13 pairs to event B and 6 pairs to event C.

p(A) = 21/36 ; p(B) = 9/36 and p(C) = 6/36.

b. Do these three events form a partition of the sample space? 0.75 pt These events do not overlap, they are mutually exclusive: no pair, among the 36, is part of two of them.

Moreover, the union of A, B and C represents the 36 pairs.

They thus form a partition of the sample space.

2) When a game is played, getting event A gives the player no winnings; event B gives him a €1 win and event C a €3 win.

a. Give the probability distribution of the random variable X : « gain of one game ». 0.5 pt

gain (€) xi 0 1 3

probability pi 21/36 9/36 6/36

b. To play a game, a player must bet one euro. On a large number of games, who wins? The organizer or

the players? 1 pt

Gain’s expectation of one game: ^E

( )

(

)

36 36

X = × + × + × = = .

Since we bet €1 on each trial, we expect to lose an average of €0.25 per trial in the long run.

For example: after 1000 trials, players will have lost an overall sum close to €250.

In conclusion: you should not play this game in the long term.

c. The game organizer expects a total of 30,000 games to be played per month, once the game has been online for some time. Give a 95% confidence interval of the monthly income that the organizer can

predict. 2 pts

Standard deviation of the gain, per game: ^σ

( )

Confidence interval of the global gain:

[ ]

0.25 30000 1.96 1.089725 30000 ; 0.25 30000 1.96 1.089725 30000 7130 ; 7870

I = × − × × × + × × =

There is a 95% probability that his monthly gain will be between €7130 and €7870.

____________________ TEST END ____________________