Problem-solving Pack

Mathematics

Year 11 · 11.2 Probability

Problem-solving Pack

Name: _________________________________

Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12

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

2Problem 2 of 12

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

3Problem 3 of 12

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

4Problem 4 of 12

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

5Problem 5 of 12

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

6Problem 6 of 12

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

7Problem 7 of 12

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

8Problem 8 of 12

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

9Problem 9 of 12

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

10Problem 10 of 12

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

11Problem 11 of 12

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

12Problem 12 of 12

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