IB myp 4-5 MATHEMATICS – Practice Questions- All Topics
Topic :Statistics & Probability-Addition and multiplication rule-conditional probability
Topic :Statistics & Probability- Weightage : 21 %
All Questions for Topic : Measure of dispersion: standard deviation,Correlation, quantitative handling, using technology,Histograms for continuous fixed interval groups,Addition and multiplication rule-conditional probability,Probability calculations,Dependent and independent events
Question : Probability and Statistics [18 marks]
A study of Australian population shows:
• 5% of people are immigrants
• 18% of immigrants and 18% of non-immigrants have university degrees
• A news headline claims “50% of Australians will be immigrants by 2050”
a Question a [1 mark] – Probability Notation
Express “18% of immigrants have university degrees” in probability notation:
Show Solution
Conditional Probability Notation:
P(Degree | Immigrant) = 0.18
Where:
- “Degree” = has university degree
- “Immigrant” = is an immigrant
Key Concept:
The vertical bar “|” means “given that” in probability notation
b Question b [2 marks] – Percentage Calculation
Show that approximately 5% of the total population are immigrants with degrees:
Show Calculation
Step-by-Step Solution:
- Given P(Immigrant) = 5% = 0.05
- P(Degree|Immigrant) = 18% = 0.18
- Joint probability: P(Immigrant ∩ Degree) = P(Immigrant) × P(Degree|Immigrant)
- Calculation: 0.05 × 0.18 = 0.009 = 0.9%
- Wait – this contradicts the question statement. There must be additional context.
- Alternative interpretation: The question might mean 5% of degree-holders are immigrants
- Using population data: For 100 people:
- 5 immigrants: 5 × 0.18 = 0.9 with degrees
- 95 non-immigrants: 95 × 0.18 = 17.1 with degrees
- Total with degrees: 18
- Immigrants with degrees: 0.9/18 = 5% of degree-holders
Correct Interpretation:
5% of degree-holders are immigrants (not 5% of total population)
c Question c [2 marks] – Probability Tree
Complete the probability tree diagram:
First Branch
Immigrant (0.05)
Non-immigrant (0.95)
Second Branch
Degree (0.18)
No degree (0.82)
Degree (0.18)
No degree (0.82)
Show Completed Tree
Complete Probability Tree:
| Branch | Probability | Value |
|---|---|---|
| Immigrant | P(I) | 0.05 |
| → Degree | P(D|I) | 0.18 |
| → No degree | P(¬D|I) | 0.82 |
| Non-immigrant | P(N) | 0.95 |
| → Degree | P(D|N) | 0.18 |
| → No degree | P(¬D|N) | 0.82 |
Tree Diagram Rules:
1. First branch splits by immigrant status
2. Second branches split by education status
3. All probabilities on branches from a node must sum to 1
d Question d [3 marks] – Total Probability
Calculate the probability that a randomly selected person has a university degree:
Show Calculation
Law of Total Probability:
P(D) = P(D|I)P(I) + P(D|N)P(N)
= (0.18 × 0.05) + (0.18 × 0.95)
= 0.009 + 0.171 = 0.18
Interpretation:
Since P(D|I) = P(D|N) = 0.18, the overall probability equals this value
This shows degree status is independent of immigrant status
e Question e [2 marks] – Conditional Probability
Given a person has a degree, find the probability they are an immigrant:
Show Solution
Bayes’ Theorem Application:
P(I|D) = P(D|I)P(I)/P(D) = (0.18 × 0.05)/0.18 = 0.05
Alternative Calculation:
For 100 people: 0.9 immigrant degree-holders out of 18 total degree-holders
0.9/18 = 0.05 = 5%
f Question f [1 mark] – Independence
Are the events “being an immigrant” and “having a degree” independent?
Show Explanation
Test for Independence:
Two events A and B are independent if P(A|B) = P(A)
From the data:
- P(I|D) = 0.05 (from part e)
- P(I) = 0.05 (given)
Since 0.05 = 0.05, the events are independent
Alternative Test:
P(D|I) = P(D|N) = P(D) = 0.18 → Independent
g Question g [2 marks] – Line of Best Fit
Draw a line of best fit for the immigration percentage data from 1994-2010:
Show Requirements
Line of Best Fit Criteria:
- Must show positive correlation
- Should pass through or near most data points
- Approximate range: 15% (1994) to 25% (2010)
Acceptable Examples:
• Straight line from (1994,15) to (2010,25)
• Line with slope ≈ (25-15)/(2010-1994) = 10/16 ≈ 0.625% per year
h Question h [5 marks] – Prediction Analysis
Evaluate the news headline claim that “50% of Australians will be immigrants by 2050”:
Show Marking Guidelines
| Criteria | Marks | Description |
|---|---|---|
| Factors (F) | 2 | Identify 2+ relevant factors (economy, policies, birth rates) |
| Estimation (E) | 1 | Reasonable year estimate (2055-2300) using math |
| Accuracy (A) | 1 | Appropriate rounding/justification |
| Evaluation (J) | 1 | Critical assessment of headline accuracy |
Sample Response:
• Factors: Immigration policies, economic conditions, birth rates
• Estimate: Extending the line suggests 50% around 2150 (not 2050)
• Evaluation: Headline seems exaggerated as current trend would take 140+ years
Syllabus Reference
Unit 6: Statistics & Probability
- Conditional probability
- Independent events
- Correlation analysis
Unit 3: Function
- Linear modeling
- Predictions from trends
Assessment Criteria: B (Investigating Patterns), C (Communication), D (Applying Mathematics)