Chapter 5, Problem 5.30P

An image is partitioned into two regions, one white and the other black. A reading taken from a randomly chosen point in the white section will be normally distributed with $\mu =4$ and ${\sigma }^2=4$, whereas one taken from a randomly chosen point in the black region will have a normally distributed reading with parameters (6, 9). A point is randomly chosen on the image and has a reading of 5. If the fraction of the image that is black is $\alpha$, for what value of $\alpha$ would the probability of making an error be the same, regardless of whether one concluded that the point was in the black region or in the white region?