Probability
Bayes Theorem
Grade 12
Question:
<p>A doctor is called to see a sick child. The doctor knows (prior to the visit) that 90% of the sick children in that neighborhood are sick with the flu, denoted by \(F\), while 10% are sick with the measles, denoted by \(M\). A well-known symptom of measles is a rash, denoted by \(R\). The probability of having a rash for a child sick with the measles is 0.95. However, occasionally children with the flu also develop a rash, with conditional probability 0.08. Upon examination the child, the doctor finds a rash. Then what is the probability that the child has the measles?</p>
<p>(1) 91/165</p>
<p>(2) 90/163</p>
<p>(3) 82/161</p>
<p>(4) 95/167</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem to find P(M|R) by computing the posterior probability: the child's observed symptom (rash) updates our prior beliefs about which disease they have.
<p><strong>Step 1: Identify given information</strong></p><p>P(F) = 0.90, P(M) = 0.10</p><p>P(R|M) = 0.95 (rash given measles)</p><p>P(R|F) = 0.08 (rash given flu)</p><p></p><p><strong>Step 2: Apply Bayes' Theorem</strong></p><p>P(M|R) = P(R|M) × P(M) / P(R)</p><p></p><p><strong>Step 3: Calculate total probability of rash</strong></p><p>P(R) = P(R|M) × P(M) + P(R|F) × P(F)</p><p>P(R) = 0.95 × 0.10 + 0.08 × 0.90</p><p>P(R) = 0.095 + 0.072</p><p>P(R) = 0.167</p><p></p><p><strong>Step 4: Calculate posterior probability</strong></p><p>P(M|R) = (0.95 × 0.10) / 0.167</p><p>P(M|R) = 0.095 / 0.167</p><p>P(M|R) = 95/167</p><p></p><p><strong>Step 5: Convert to decimal and simplify if needed</strong></p><p>P(M|R) ≈ 0.569 or 95/167 ≈ 57%</p><p></p><p>∴ Answer: The probability that the child has measles is <strong>95/167 ≈ 0.569 or approximately 57%</strong></p>
Correct Answer: 4