Bayes’ Theorem: Flipping the Question
Bayes’ Theorem: Find \(P(A| B)\) using \(P(B | A)\).
- Formula: \(P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)}\).
- Total Probability: \(P(B) = \sum_{i} P(B | A_i) \cdot P(A_i)\) for partitions \(A_i\).
Example: Pregnancy test (Exercise #1)
- \(P(\text{Pregnant}) = 0.1\), \(P(\text{Positive | Pregnant}) = 0.98\).
- Bayes’ finds: \(P(\text{Pregnant | Positive})\).
Exercise #1
If a woman takes an early pregnancy test, she will either test positive, meaning that the test says she is pregnant, or test negative, meaning that the test says she is not pregnant. Suppose that if a woman really is pregnant, there is a 98% chance that she will test positive. Also suppose that if a woman really is not pregnant, there is a 99% chance that she will test negative.
Suppose that 1,000 woman take early pregnancy tests and that 100 of them truly pregnant. What is the probability that a randomly chosen woman from this group will test positive?
Suppose a woman tests positive. What is the probability that she really is pregnant?
Overview
This analysis uses a sample of 1,000 women undergoing early pregnancy tests, with the following assumptions:
- If a woman is truly pregnant, the test is positive 98% of the time.
- If a woman is not pregnant, the test is negative 99% of the time.
Total Population:
- 1,000 women, with 100 truly pregnant (10%) and 900 not pregnant (90%).
Probability Calculations:
- Probability of a Positive Test (Overall) \[ P(\text{Test Positive}) = P(\text{Test Positive | Pregnant}) \times P(\text{Pregnant}) + P(\text{Test Positive | Not Pregnant}) \times P(\text{Not Pregnant}) \]
\[ = 0.98 \times 0.1 + 0.01 \times 0.9 = 0.107 \] - Result: 10.7% chance of testing positive across all 1,000 women.
- Probability of Being Pregnant Given a Positive Test
\[ P(\text{Pregnant | Test Positive}) = \frac{P(\text{Test Positive | Pregnant}) \times P(\text{Pregnant})}{P(\text{Test Positive})} \]
\[ = \frac{0.98 \times 0.1}{0.107} \approx 0.916 \] - Result: 91.6% chance of being pregnant if tested positive.
Tree Diagram for Probability Calculation
This diagram illustrates the probability pathways based on pregnancy status and test outcomes:
graph TD;
A[Start: 1,000 Women] -->|10% Pregnant| B[Pregnant]
A -->|90% Not Pregnant| C[Not Pregnant]
B -->|98% Test Positive| D[Test Positive]
B -->|2% Test Negative| E[Test Negative]
C -->|1% Test Positive| F[Test Positive]
C -->|99% Test Negative| G[Test Negative]
style D fill:#4daf4a,color:#ffff
style E fill:#e41a1c,color:#ffff
style F fill:#4daf4a,color:#ffff
style G fill:#4daf4a,color:#ffff
