Probability
Bayes Theorem
Grade 12
Question:
<p>On a Saturday night, 20% of all drivers in U.S.A. are under the influence of alcohol. The probability that a driver under the influence of alcohol will have an accident is 0.001. The probability that a sober driver will have an accident is 0.0001. If a car on a Saturday night smashed into a tree, the probability that the driver was under the influence of alcohol is</p>
<p>(1) 3/7</p>
<p>(2) 4/7</p>
<p>(3) 5/7</p>
<p>(4) 6/7</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem to find the posterior probability that a driver was drunk given that an accident occurred. We need P(Drunk|Accident) = P(Accident|Drunk)·P(Drunk) / P(Accident), where P(Accident) is found using the law of total probability.
<p><strong>Step 1:</strong> Define events and given data:</p><p>Let D = driver is drunk, A = accident occurs</p><p>P(D) = 0.2, P(D') = 0.8</p><p>P(A|D) = 0.001, P(A|D') = 0.0001</p><p></p><p><strong>Step 2:</strong> Find total probability of accident using law of total probability:</p><p>P(A) = P(A|D)·P(D) + P(A|D')·P(D')</p><p>P(A) = 0.001 × 0.2 + 0.0001 × 0.8</p><p>P(A) = 0.0002 + 0.00008 = 0.00028</p><p></p><p><strong>Step 3:</strong> Apply Bayes' theorem to find P(D|A):</p><p>P(D|A) = P(A|D)·P(D) / P(A)</p><p>P(D|A) = (0.001 × 0.2) / 0.00028</p><p>P(D|A) = 0.0002 / 0.00028</p><p>P(D|A) = 200/280 = 5/7 ≈ 0.714</p><p></p><p>∴ Answer: 5/7 or approximately 0.714 (or 71.4%)</p>
Correct Answer: 2