Margaret is a 65-year-old lady who had a mammography. Mammography is used for screening for breast cancer. Her doctor told her that the test was negative and that mammography has 9... Margaret is a 65-year-old lady who had a mammography. Mammography is used for screening for breast cancer. Her doctor told her that the test was negative and that mammography has 90% sensitivity, 80% specificity, and that 3 in 10 women of her age have breast cancer. How likely is it that Margaret is indeed free of breast cancer?
Understand the Problem
The question is asking us to determine the probability that Margaret does not have breast cancer given the results of her mammography test, considering the sensitivity, specificity, and prevalence of breast cancer in her age group.
Answer
$0.9205$
Answer for screen readers
The probability that Margaret does not have breast cancer given a positive mammography test is approximately $0.9205$.
Steps to Solve
- Identify Given Information
We have the following data:
- Sensitivity (True Positive Rate) = P(Positive Test | Disease) = 0.85
- Specificity (True Negative Rate) = P(Negative Test | No Disease) = 0.90
- Prevalence = P(Disease) = 0.01
- Calculate the Probability of No Disease
The probability of not having the disease can be calculated using: $$ P(No Disease) = 1 - P(Disease) = 1 - 0.01 = 0.99 $$
- Calculate the Probability of a Positive Test Given No Disease
Using specificity, we find the probability that a person tests positive given that they do not have the disease: $$ P(Positive Test | No Disease) = 1 - P(Negative Test | No Disease) = 1 - 0.90 = 0.10 $$
- Apply Bayes' Theorem to Find P(No Disease | Positive Test)
Now we apply Bayes' theorem: $$ P(No Disease | Positive Test) = \frac{P(Positive Test | No Disease) \cdot P(No Disease)}{P(Positive Test)} $$
- Calculate the Overall Probability of a Positive Test
To calculate $P(Positive Test)$, we consider both cases (having and not having the disease): $$ P(Positive Test) = P(Positive Test | Disease) \cdot P(Disease) + P(Positive Test | No Disease) \cdot P(No Disease) $$ Substituting the values: $$ P(Positive Test) = (0.85 \cdot 0.01) + (0.10 \cdot 0.99) $$ Calculating this gives: $$ P(Positive Test) = 0.0085 + 0.099 = 0.1075 $$
- Calculate P(No Disease | Positive Test)
Substituting back into Bayes' theorem: $$ P(No Disease | Positive Test) = \frac{0.10 \cdot 0.99}{0.1075} $$ Calculating this: $$ P(No Disease | Positive Test) = \frac{0.099}{0.1075} \approx 0.9205 $$
The probability that Margaret does not have breast cancer given a positive mammography test is approximately $0.9205$.
More Information
This probability indicates that even with a positive test, there is still a high chance that Margaret does not have breast cancer, highlighting the importance of understanding sensitivity, specificity, and prevalence in medical testing.
Tips
- Confusing sensitivity with specificity; remember that sensitivity relates to detecting disease correctly, whereas specificity relates to correctly identifying those without the disease.
- Not calculating the overall probability of a positive test properly; ensure all components are considered.
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