Worked Solution: IB Math AA HL 2024 Paper 1 TZ1 Q1 (Fully Explained)
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🎲 Probability Problem: Dice Rolls and Frequency
This article presents a fully explained solution to an IB-style probability question involving frequency tables and statistical calculations.
Problem Statement (Total Marks: 6)
Claire rolls a standard six-sided die 16 times. The observed outcomes are shown below:
| Outcome | Frequency |
|---|---|
| 1 | p |
| 2 | q |
| 3 | 4 |
| 4 | 2 |
| 5 | 0 |
| 6 | 3 |
The mean of the 16 results is 3.
Part (b): If each result is multiplied by 10, what is the new average score? [1 mark]
Solution
Step 1: Total number of rolls
p + q + 4 + 2 + 0 + 3 = 16
⇒ p + q = 7 …(1)
Step 2: Compute the sum of all outcomes
Sum = 1p + 2q + 3*4 + 4*2 + 5*0 + 6*3
= p + 2q + 12 + 8 + 0 + 18
= p + 2q + 38
Step 3: Use the given mean
Mean = (Sum of outcomes) / (Total rolls)
3 = (p + 2q + 38) / 16
⇒ p + 2q + 38 = 48
⇒ p + 2q = 10 …(2)
Step 4: Solve the system of equations
From (1): p + q = 7
From (2): p + 2q = 10
Subtracting (1) from (2):
(p + 2q) – (p + q) = 10 – 7
⇒ q = 3
Substitute back into (1):
p + 3 = 7
⇒ p = 4
Final Answer:
p = 4, q = 3
Part (b): New Mean When Values Are Multiplied by 10
If each outcome is multiplied by 10, the mean is also multiplied by 10.
New Mean = 10 × 3 = 30
Answer: 30
🔁 Alternative Approaches
Part (a)
Use total frequency = 16
Mean = (Sum of (Outcome × Frequency)) / Total Frequency = 3
Set up the same equations and solve.
Part (b)
Multiplying each value by a constant also multiplies the mean by that constant.
So, new mean = 10 × original mean
📌 Marking Criteria – IB Math AA HL 2024 P1 TZ1 Q1
(a) [5 marks]
- M1: Attempts equation from total freq = 16 or mean = 3
- A1: Forms correct equation:
p + q = 7 - A1: Uses mean:
(p + 2q + 38)/16 = 3 ⟹ p + 2q = 10 - M1: Solves system: e.g.,
p + 2q = 10,p + q = 7 - A1: Final correct values:
p = 4,q = 3Note: Direct answer without working gets M1 A0 A0 M0 A1
(b) [1 mark]
- A1: New mean =
3 × 10 = 30
Total: 6 marks
🧠 Key Strategy: Solving Frequency and Mean Problems
- Use total rolls and given mean to create equations
- Solve the system of equations using substitution or elimination
- Multiply means by constants directly if data is scaled
- Double-check your calculations
🚫 Common Mistakes to Avoid
- Forgetting a term in the sum
- Wrong total frequency
- Arithmetic mistakes in solving equations
- Misapplying mean formula
- Getting negative frequencies (not possible!)
🧠 Expert Insights
- Mean is affected by both origin and scale.
- Mean, also known as Arithmetic Mean (AM) or the Expectation E(x).
- Find the difference between AM, GM, HM, and Trimmed Mean.
📚 Practice Problems
Practice Problem 1
A die is rolled 20 times. The frequencies are:
| Outcome | Frequency |
|---|---|
| 1 | a |
| 2 | b |
| 3 | 5 |
| 4 | 3 |
| 5 | 2 |
| 6 | 1 |
Given mean = 2.75. Find a and b.
Step 1
a + b + 5 + 3 + 2 + 1 = 20 ⇒ a + b = 9 …(1)
Step 2
Sum = 1a + 2b + 3*5 + 4*3 + 5*2 + 6*1
= a + 2b + 15 + 12 + 10 + 6 = a + 2b + 43
Mean = (a + 2b + 43) / 20 = 2.75
⇒ a + 2b + 43 = 55 ⇒ a + 2b = 12 …(2)
Solving:
(2) – (1): (a + 2b) – (a + b) = 12 – 9
⇒ b = 3
⇒ a = 6
Answer: a = 6, b = 3
Practice Problem 2
What is the new mean if every outcome is multiplied by 5?
New mean = 5 × 2.75 = 13.75
🔬 Other Challenge
Advanced Problem 1
A die is rolled 18 times:
| Outcome | Frequency |
|---|---|
| 1 | x |
| 2 | y |
| 3 | 3 |
| 4 | 4 |
| 5 | 1 |
| 6 | 2 |
Mean is 3. Find x, y and the new mean if each result is doubled.
Step 1
x + y + 3 + 4 + 1 + 2 = 18 ⇒ x + y = 8 …(1)
Step 2
Sum = 1*x + 2*y + 3*3 + 4*4 + 5*1 + 6*2
= x + 2*y + 9 + 16 + 5 + 12 = x + 2y + 42
Mean = (x + 2y + 42) / 18 = 3
⇒ x + 2y + 42 = 54 ⇒ x + 2y = 12 …(2)
Solving:
(2) – (1): (x + 2y) – (x + y) = 12 – 8 ⇒ y = 4
⇒ x = 4
New Mean if values are doubled:
2 × 3 = 6
Advanced Problem 2
Find the variance of the original results.
Full working steps are in the downloadable PDF.
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