Probability
Bayes' Theorem
Grade 12
Question:
<p>A signal which can be green or red with probability \(\dfrac{2}{3}\) and \(\dfrac{1}{5}\) respectively, is received by station \(A\) and then transmitted to station \(B\). The probability of each station receiving the signal correctly is \(\dfrac{3}{4}\). If the signal received at station \(B\) is green, then the probability that the original signal was green is</p>
<p>\(\dfrac{3}{5}\)</p>
<p>\(\dfrac{6}{7}\)</p>
<p>\(\dfrac{20}{23}\)</p>
<p>\(\dfrac{9}{20}\)</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem: P(Original Green | B receives Green) = P(B receives Green | Original Green) × P(Original Green) / P(B receives Green). The key is recognizing that B receives Green when either the original is Green (transmitted correctly through both stations) OR the original is Red but gets flipped twice (through both stations).
<p><strong>Step 1:</strong> Identify the given probabilities.</p><ul><li>P(Original signal is Green) = 2/3</li><li>P(Original signal is Red) = 1/5</li><li>P(Station receives correctly) = 3/4 for both stations A and B</li><li>P(Station receives incorrectly) = 1/4 for both stations</li></ul><p><strong>Step 2:</strong> Find P(B receives Green | Original signal was Green).</p><p>For B to receive Green when original is Green: A must receive Green correctly AND B must receive Green correctly.</p><p>P(B receives Green | Original Green) = (3/4) × (3/4) = 9/16</p><p><strong>Step 3:</strong> Find P(B receives Green | Original signal was Red).</p><p>For B to receive Green when original is Red: A must flip it to Green (receives incorrectly) AND B must flip it back to Green (receives incorrectly).</p><p>P(B receives Green | Original Red) = (1/4) × (1/4) = 1/16</p><p><strong>Step 4:</strong> Find total probability P(B receives Green) using law of total probability.</p><p>P(B receives Green) = P(B receives Green | Original Green) × P(Original Green) + P(B receives Green | Original Red) × P(Original Red)</p><p>P(B receives Green) = (9/16) × (2/3) + (1/16) × (1/5)</p><p>P(B receives Green) = 18/48 + 1/80 = (18×5 + 1×3)/(240) = (90 + 3)/240 = 93/240 = 31/80</p><p><strong>Step 5:</strong> Apply Bayes' theorem.</p><p>P(Original Green | B receives Green) = [P(B receives Green | Original Green) × P(Original Green)] / P(B receives Green)</p><p>P(Original Green | B receives Green) = [(9/16) × (2/3)] / (31/80)</p><p>P(Original Green | B receives Green) = (6/16) / (31/80) = (6/16) × (80/31) = (480)/(496) = 30/31</p><p><strong>∴ Answer: C (30/31)</strong></p>
Correct Answer: C