A student is going to a party but is unsure whether it's overalls or regular dress that's the norm. He therefore assumes that the proportion of party attendees wearing overalls, $\Theta$, is uniformly distributed over $[0,1]$. On his way home to change, he sees another student heading to the party dressed in overalls. He now faces a choice: to wear his own overall, $\left(a_1\right)$, or not, $\left(a_2\right)$. He doesn't want to stand out, so if he's in the majority of students at the party (in terms of dress), his loss function is zero. However, he thinks it's four times as embarrassing to wear overalls if the majority of students aren't wearing them compared to the other way around. Calculate the ratio between the posterior risk for $a_1$ and $a_2$ and determine the optimal choice. Suppose now he sees another student dressed in overalls heading to the party. How does this affect his decision? (Assume this is a large party.)
$\theta$ is uniformly distributed, I assume I should look for the data distribution? The answer is $\frac{4}{7}$