False Positives and False Negatives (Grade 12 NSC Matric Mathematical Literacy): Revision Notes
False Positives and False Negatives
Understanding test results and their reliability
Medical tests, pregnancy tests, drug tests, and other screening procedures are generally very accurate, but there is always a small chance that results might be incorrect. Understanding when and why tests can be wrong helps us make better decisions about the information we receive.
When dealing with any test that gives a "Yes" or "No" answer, you need to consider that the test could be wrong in two different ways:
- The test could be wrong when it gives a positive result (says "Yes" incorrectly)
- The test could be wrong when it gives a negative result (says "No" incorrectly)
These incorrect results have significant implications in real-world situations, affecting medical decisions, athletic careers, and personal well-being.
Key definitions
False positive: This occurs when a test says "Yes" but the real situation is actually "No". The test gives a positive result when it should be negative.
False negative: This occurs when a test says "No" but the real situation is actually "Yes". The test gives a negative result when it should be positive.
Test accuracy: The percentage of correct results a test produces across all cases tested.
Confusion matrix: A 2×2 table that shows all possible outcomes when comparing test results to reality.
These incorrect results have important consequences in real life. A false positive in medical testing might cause unnecessary worry and further testing, while a false negative might mean missing a condition that needs treatment.
The confusion matrix
To understand how tests can be right or wrong, we use a simple table called a confusion matrix. This shows all four possible outcomes when comparing test results to reality.
Common mistake to avoid: Don't confuse which quadrant represents false positives versus false negatives. Remember that false positive = test says "Yes" when reality is "No".
The table shows that:
- When the real situation is "Yes" and the test says "Yes" → The test is correct (True Positive)
- When the real situation is "Yes" and the test says "No" → False negative
- When the real situation is "No" and the test says "Yes" → False positive
- When the real situation is "No" and the test says "No" → The test is correct (True Negative)
Remember that incorrect results are usually very unlikely, but they do happen and we need to account for them when interpreting test results.
Understanding test accuracy
When we say a test is "95% accurate", this means that 95% of the time the test gives the correct result. However, this also means that 5% of the time (1 in 20 tests) the result will be wrong - either a false positive or false negative.
The key insight is that even highly accurate tests will produce some incorrect results when used on large populations. This is why medical professionals often require confirmatory testing when initial results are positive.
Even with our best testing methods, we can only provide the best estimate with available information - perfect certainty is mathematically impossible in probability.
Worked example: Virus testing
From this data we can see:
- True positives: 4,885 people correctly identified as carrying the virus
- False negatives: 115 people who carry the virus but tested negative
- False positives: 2,630 people who don't carry the virus but tested positive
- True negatives: 932,370 people correctly identified as not carrying the virus
Key calculations from the example
Probability that a person without the virus tests positive:
- Number of false positives = 2,630
- Total people without virus = 995,000
- Probability =
This demonstrates why expensive laboratory confirmation tests are used when quick tests show positive results - to reduce the impact of false positives.
Real-world application: Drug testing in sports
Real-World Case Study: Comrades Marathon Drug Testing
The Comrades Marathon case study provides an excellent example of how false positives affect people's lives and why multiple testing is crucial.
The Scenario: In 2013, runner Ludwick Mamabola initially tested positive for methylhexaneamine in his B-sample drug test. However, after expert analysis, he was later cleared of doping charges when the testing process was found to be inadequate.
Understanding the probabilities: If a drug test has 95% accuracy:
- Probability of false positive = 5% or 0.05
- Probability of false negative = 5% or 0.05
Key calculation: If the first test gives a false positive, what are the chances the second test is also wrong?
- Each test is independent
- Probability of both tests being wrong =
- This is why athletes can request B-sample testing when A-samples test positive
We cannot be 100% certain about any test result because there always exists a small possibility of error, even with multiple tests. However, the probability of error becomes extremely small with proper confirmatory testing.
Exam tips and common mistakes
Critical Points for Exam Success:
- Don't confuse terminology: False positive means the test incorrectly says "Yes", false negative means the test incorrectly says "No"
- Accuracy works both ways: A 95% accurate test can have both false positives AND false negatives
- Population size matters: Even small error rates become significant when testing millions of people
- Multiple testing reduces uncertainty: This is why confirmatory tests are used in medical and sports contexts
- Perfect certainty is impossible: Probability gives us the best estimate with available information, but some uncertainty always remains
Practical problem-solving approach
Step-by-Step Problem Solving Method:
When solving false positive/negative problems, follow these steps:
- Identify what the test is checking for
- Determine the accuracy rate or error rate
- Calculate expected false positives/negatives using proportions
- Consider the population size - larger populations will have more false results even with accurate tests
- Remember conversion: Percentages need to be converted to decimals for calculations
Example conversion: 0.4% = 0.004 =
Summary
Key Points to Remember:
- False positive: Test says "Yes" when the real answer is "No"
- False negative: Test says "No" when the real answer is "Yes"
- Even accurate tests make mistakes - this is why multiple tests are often required
- Large populations amplify small error rates - 0.4% error seems small, but affects thousands when testing millions
- Perfect certainty is impossible - probability helps us make the best decisions with available information
- Mathematical relationships: Use proportions and basic multiplication to calculate expected false results in populations