Question:
We have two coins. One is biased with probability of head = 1.0 and the other is a fair coin.
One coin is chosen at random and tossed twice.
If both outcomes are heads, the probability that the chosen coin is fair is ____________ (correct to one decimal place).
We have two coins. One is biased with probability of head = 1.0 and the other is a fair coin.
One coin is chosen at random and tossed twice.
If both outcomes are heads, the probability that the chosen coin is fair is ____________ (correct to one decimal place).
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Bayes’ theorem updates prior probabilities using observed evidence.
Updated On: Dec 2, 2025
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Correct Answer: 0.2
Solution and Explanation
Let \(F\) = fair coin, \(B\) = biased coin.
\[
P(F)=P(B)=\frac{1}{2}
\]
Likelihoods:
\[
P(HH|F)=0.5^2=0.25,\qquad P(HH|B)=1
\]
Apply Bayes' theorem:
\[
P(F|HH)=\frac{P(HH|F)P(F)}{P(HH|F)P(F)+P(HH|B)P(B)}
\]
\[
=\frac{0.25\times 0.5}{0.25\times 0.5 + 1\times 0.5}
\]
\[
= \frac{0.125}{0.625} = 0.2
\]
Thus, the answer lies in:
\[
\boxed{0.2\ \text{to}\ 0.2}
\]
Final Answer: 0.2
Final Answer: 0.2
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