In a study, physicians were asked what the odds of breast cancer would be in a woman who was initially thought to have a 1% risk of cancer but who ended up with a positive mammogram result (a mammogram accurately classifies about 80% of cancerous tumors and 90% of benign tumors.)
95 out of a hundred physicians estimated the probability of cancer to be about 75%. Do you agree?
Solution
Events
$P$ = mammogram result is positive,
$B$ = tumor is benign,
$M$ = tumor is malignant.
Given that $P(M) = 0.01$. Since $B^c = M$, we have $P(B) = 1-P(M) = 0.99$.
Also given that the conditional probabilities
$P(P|M) = 0.80$ and $P(P^c|B) =0.90$. Thus $P(P|B) =1- P(P^c|B)= 1- 0.90= 0.10$.
Using Bayes' formula, we have
$$ \begin{aligned} P(M |P) &= \frac{P(P|M)P(M)}{(P(P|M)P(M) + P(P|B)P(B))}\\ &= \frac{0.80 \times 0.01}{0.80 \times 0.01 + 0.10 \times 0.99}\\ &= 0.075 \end{aligned} $$
That is the chance would be 7.5%, which is far from a common estimate of 75%.
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
