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).

Hint:

Bayes’ theorem updates prior probabilities using observed evidence.

Answer 1

Correct Answer: 0.2

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