Probability
Bayes' Theorem
Grade 12
Question:
<p>There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. When one of the three coins is chosen at random and tossed, it showed heads. What is the probability that it was the two-headed coin?</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem: the probability of selecting the two-headed coin given that heads appeared equals P(heads|two-headed) × P(two-headed) divided by the total probability of getting heads from any coin.
<p><strong>Step 1:</strong> Identify the probabilities for each coin.</p><ul><li>Two-headed coin: P(H|Coin₁) = 1</li><li>Biased coin: P(H|Coin₂) = 0.75 = 3/4</li><li>Unbiased coin: P(H|Coin₃) = 0.5 = 1/2</li></ul><p><strong>Step 2:</strong> Each coin is chosen with equal probability: P(Coin₁) = P(Coin₂) = P(Coin₃) = 1/3</p><p><strong>Step 3:</strong> Calculate total probability of getting heads using the law of total probability:</p><p>P(H) = P(H|Coin₁)·P(Coin₁) + P(H|Coin₂)·P(Coin₂) + P(H|Coin₃)·P(Coin₃)</p><p>P(H) = 1·(1/3) + (3/4)·(1/3) + (1/2)·(1/3)</p><p>P(H) = 1/3 + 3/12 + 1/6 = 4/12 + 3/12 + 2/12 = 9/12 = 3/4</p><p><strong>Step 4:</strong> Apply Bayes' theorem to find P(Coin₁|H):</p><p>P(Coin₁|H) = [P(H|Coin₁)·P(Coin₁)] / P(H)</p><p>P(Coin₁|H) = [1·(1/3)] / (3/4) = (1/3) × (4/3) = 4/9</p><p><strong>∴ Answer: 4/9</strong></p>
Correct Answer: 4/9