Bayes Theorem

We are given 19 unbiased coins and 1 biased coin. The unbiased coin has a probability of of showing heads, while the biased coin always shows heads. We are asked to find the probability that an unbiased coin was selected given that the head appeared on the toss. Let: - be the event that an unbiased coin was selected. - be the event that the biased coin was selected. - be the event that a head is tossed. We are looking for , the probability that an unbiased coin was selected given that the head appeared. By Bayes' theorem: ### Step 1: Calculate The probability of getting heads given that an unbiased coin was selected is . ### Step 2: Calculate The probability of selecting an unbiased coin is . ### Step 3: Calculate The probability of getting heads given that the biased coin was selected is . ### Step 4: Calculate The probability of selecting the biased coin is . ### Step 5: Calculate The total probability of getting heads is: ### Step 6: Apply Bayes' theorem Now, applying Bayes' theorem: Thus, the correct answer is .