Résolution de problèmes — Sport – Conditional Probability and Decision-Making

1Problem 1 of 24

Universal set operations. The universal set is

U = \{1, 2, 3, \ldots, 15\}

. Let

A = \{x \in U : x \text{ is a multiple of } 3\}

and

B = \{x \in U : x \text{ is a factor of } 12\}

.

(a) List the elements of

A

,

B

,

A \cap B

,

A \cup B

, and

A'

.
(b) State whether

A

and

B

are mutually exclusive. Justify.
(c) Find

n(A' \cap B)

and

n(A \cup B)'

.

Working space

2Problem 2 of 24

Venn diagram from sentence. In a class of 30 students, 18 play tennis, 12 play hockey, and 5 play both.

(a) Draw a Venn diagram with all four regions labelled.
(b) How many play neither tennis nor hockey?
(c) Find

P(\text{plays tennis only})

.
(d) Find

n(T \cup H)

and

n((T \cup H)')

.

Working space

3Problem 3 of 24

Three-set Venn — school subjects. At a school of 100 students: 55 enjoy Maths (M), 48 enjoy Science (S), 30 enjoy Art (A). 22 enjoy M & S, 10 enjoy S & A, 15 enjoy M & A, and 6 enjoy all three.

(a) Use the inclusion–exclusion principle to find

n(M \cup S \cup A)

.
(b) Find the number who enjoy exactly one of the three subjects.
(c) Find the probability that a randomly chosen student enjoys none of the three.

Working space

4Problem 4 of 24

Languages survey. In a class of 25 students, 15 study French, 12 study Spanish, and 4 study neither.

(a) How many study at least one language?
(b) Use the inclusion–exclusion formula to find the number who study both.
(c) Draw a Venn diagram and fill in all four regions.

Working space

5Problem 5 of 24

Sample space — two dice. Two fair six-sided dice are rolled.

(a) State the size of the sample space.
(b) List the outcomes in the event

E

= "the sum is 7".
(c) List the outcomes in the event

F

= "at least one die shows a 6".
(d) Find

n(E \cap F)

and

n(E \cup F)

.

Working space

6Problem 6 of 24

Three-set Venn — fitness app. A fitness app tracks three habits among 200 users: running (R), cycling (C), swimming (S).

- 80 run, 70 cycle, 60 swim.
- 30 run and cycle, 25 run and swim, 20 cycle and swim.
- 10 do all three.

(a) Find the number who do at least one of the three activities.
(b) Find the number who do none.
(c) Find the number who do exactly two.

Working space

7Problem 7 of 24

Algebraic Venn fill. In a 2-set Venn diagram, "

A

only" contains

2x

elements, "

B

only" contains

x + 5

, "both" contains

x

, and "neither" contains 4 elements. The universal set has 25 elements.

(a) Set up an equation in

x

.
(b) Solve for

x

.
(c) State

n(A)

,

n(B)

,

n(A \cap B)

, and

n(A \cup B)

.

Working space

8Problem 8 of 24

De Morgan in action [EXT]. Let

U = \{1, 2, \ldots, 10\}

,

A = \{2, 3, 5, 7\}

,

B = \{2, 4, 6, 8, 10\}

.

(a) Find

A \cup B

and

(A \cup B)'

.
(b) Find

A'

and

B'

and hence find

A' \cap B'

.
(c) Verify your answer to (a) and (b) match De Morgan's law

(A \cup B)' = A' \cap B'

.

Working space

9Problem 9 of 24

Lifting from a conditional context. In a survey,

n(U) = 50

,

n(A) = 24

,

n(B) = 20

,

n(A \cap B) = 12

.

(a) Find

n(A \cup B)

and

n(A \cup B)'

.
(b) Find

n(A \cap B')

and

n(A' \cap B)

.
(c) Construct a 2-set Venn diagram showing all four regions.

Working space

10Problem 10 of 24

Set-builder & interval. Express each set in interval notation; then describe in words.

(a)

\{x \in \mathbb{R} : -2 \leq x < 5\}

(b)

\{x \in \mathbb{R} : x > 3\}

(c)

\{x \in \mathbb{R} : 0 \leq x \leq 10 \text{ and } x \neq 5\}

Working space

11Problem 11 of 24

Subsets and power set [EXT]. Let

A = \{a, b, c, d\}

.

(a) How many subsets does

A

have?
(b) List the subsets of size 2.
(c) Explain in words why a set with

n

elements has

2^n

subsets.

Working space

12Problem 12 of 24

Modelling — sports club membership. A sports club has 100 members. Each plays at least one of football (F), tennis (T), or swimming (S). 60 play F, 50 play T, 40 play S. 20 play F & T, 15 play F & S, 10 play T & S.

x

members play all three.

(a) Show that

x = 5

.
(b) How many members play exactly one sport?
(c) The treasurer wants to send a discount voucher to members who play more than one sport. How many vouchers are needed?

Working space

13Problem 13 of 24

Die events. A fair die is rolled. Let

A

= "score is even" and

B

= "score is greater than 3".

(a) Find

P(A)

,

P(B)

,

P(A \cap B)

and

P(A \cup B)

.
(b) Are

A

and

B

independent? Justify.

Working space

14Problem 14 of 24

Tree with replacement. A bag contains 4 red and 3 blue counters. Two are drawn one at a time, with replacement.

(a) Draw a tree diagram.
(b) Find

P(\text{both red})

.
(c) Find

P(\text{exactly one red})

.

Working space

15Problem 15 of 24

Without replacement. A bag contains 5 red and 4 blue marbles. Two are drawn without replacement.

(a) Draw a tree diagram showing all four paths.
(b) Find

P(\text{both blue})

.
(c) Find

P(\text{one of each colour})

.

Working space

16Problem 16 of 24

Diagnostic test (rare condition). A test is used to detect a rare condition.

- 2% of people have the condition.
- 90% of those who have it test positive.
- 5% of those who do not have it test positive (false positives).

(a) Draw a tree diagram with the four outcomes.
(b) Find

P(\text{positive test})

.
(c) Given a positive test, find

P(\text{has the condition})

. Comment briefly.

Working space

17Problem 17 of 24

Languages — independence. At a school, 40% of students study French (

F

) and 25% study Spanish (

S

). Of those who study French, 30% also study Spanish.

(a) Find

P(F \cap S)

.
(b) Find

P(F \cup S)

.
(c) Are

F

and

S

independent? Justify.

Working space

18Problem 18 of 24

Reverse conditional. Two machines,

M_1

and

M_2

, produce identical items.

M_1

makes 60% of items with defect rate 2%;

M_2

makes 40% with defect rate 5%.

(a) Find

P(\text{defective})

.
(b) Given a defective item, find

P(M_2 \mid \text{defective})

.
(c) A buyer claims "most defectives come from

M_1

because

M_1

makes more items." Critique this claim.

Working space

19Problem 19 of 24

At least one — design question. A vaccine has a 70% chance of being effective per person, treated as independent trials.

(a) Find the probability that the first 3 people all benefit.
(b) Find the smallest

n

such that

P(\text{at least one benefits}) \geq 0.999

.
(c) If 5 people are vaccinated, find

P(\text{exactly 3 benefit})

.

Working space

20Problem 20 of 24

Combinatorial code. A four-digit code is formed using the digits

1,2,3,4,5

without repetition.

(a) How many different codes are possible?
(b) Find

P(\text{code is even})

.
(c) Find

P(\text{code starts with 1 and ends with 5})

.

Working space

21Problem 21 of 24

Conditional on a Venn. In a survey of 50 people, 30 like tea, 25 like coffee, and 12 like both.

(a) Construct a 2-set Venn diagram and fill in all four regions.
(b) Find

P(\text{tea})

,

P(\text{coffee})

,

P(\text{both})

and

P(\text{neither})

.
(c) Given the person likes coffee, find

P(\text{also tea})

.
(d) Are "likes tea" and "likes coffee" independent? Justify.

Working space

22Problem 22 of 24

Mixed scenario — sport, music & both. At a youth club, every member plays sport (

S

), studies music (

M

), or both. 60% play sport, 50% study music.

(a) Show that 10% do both.
(b) Find

P(\text{plays sport} \mid \text{studies music})

.
(c) Are sport and music independent at this club? Justify.
(d) A different club has 70% sport and 40% music, genuinely independent. What percentage does both, and what percentage neither?

Working space

23Problem 23 of 24

Two-way table — phone survey. A sample of 200 students reports phone ownership and tablet ownership.

| | Tablet | No tablet | Total |
|--------------|--------|-----------|-------|
| Phone | 60 | 90 | 150 |
| No phone | 20 | 30 | 50 |
| Total | 80 | 120 | 200 |

(a) Find

P(\text{phone})

,

P(\text{tablet})

,

P(\text{phone and tablet})

.
(b) Are "phone" and "tablet" independent? Justify with a calculation.
(c) Given a student owns a tablet, find

P(\text{phone})

.

Working space

24Problem 24 of 24

Reverse conditional in context. A factory's QA pipeline classifies items as defective (D) or fine. Two product lines (X and Y) feed the same conveyor:

- 70% of items come from line X with defect rate 1%.
- 30% of items come from line Y with defect rate 5%.

(a) An item is taken at random. Find

P(D)

.
(b) Given the item is defective, find

P(Y \mid D)

.
(c) Quality control says "since most output is from X, most defectives are from X." Critique using your answers.

Working space

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