Answer

Prerequisites

• Algebraic Simplification
• Probability Inequalities

Step-by-Step Derivation

From part (a), the decision rule is to guess heads if: $(1 - \theta_1) \alpha > \theta_2 (1 - \alpha)$

Given the condition $\theta_1 = \theta_2 = \theta$, we substitute $\theta$ into the inequality: $(1 - \theta) \alpha > \theta (1 - \alpha)$

Now, we simplify the inequality: $\alpha - \theta \alpha > \theta - \theta \alpha$

Adding $\theta \alpha$ to both sides: $\alpha > \theta$

Thus, when the error rates are symmetric ($\theta_1 = \theta_2 = \theta$), the optimal decision rule simplifies to: guess heads if the prior probability of heads ($\alpha$) is strictly greater than the probability of the friend reporting incorrectly ($\theta$).