Retrieve index of random item in numpy array with a specific value

I have a 2d numpy array with a small range of different values. I want to retrieve a random index, but with a specific value. For example say this is the array:

arr = [[1,2,1,1,3],
       [4,3,4,1,1],
       [2,3,1,2,1],
       [1,2,1,1,2],
       [3,1,3,4,2]]

I want a random index of any value which is a 1, for example (2,2). The only way I have been able to do this so far is to take any random index, and then check afterwards if it contains the value I want and then redraw if it isn't. This would be ok for the above example, but I will need to retrieve many random indices from a large array.

Is there a method or algorithm for the procedure I am describing?

The application of this is to provide random pixels of a classified image to perform an accuracy assessment.