$4.$ Suppose you have a set of images where each image is either a dog or a cat, and you$$ve trained a machine learning algorithm that tries to distinguish dogs from cats. You know the three following facts:

$$ Your image set is $1/3$ cats and $2/3$ dogs.

$$ When given a picture of a dog the algorithm correctly labels it as a dog $90\%$ of the time.

$$ When given a picture of a cat the algorithm correctly labels it as a cat $80\%$ of the time.

You$$ve asked the algorithm to sort the entire set of images. Consider the set of all images which the algorithm has misidentified, i$.$e$.$ all the cat images that it labeled as dogs and all the dog images it labeled as cats. If you select one image at random from that set, what is the probability that it is an image of a dog?