The method of randomized response is sometimes used to conduct surveys on sensitive topics. A simple version of the method can be described as follows: A random sample of n persons is drawn from a large population. For each person in the sample there is probability 1/2 that the person will be asked a standard question and probability 1/2 that the person will be asked a sensitive question. Furthermore, this selection of the standard or the sensitive question is made independently from person to person. If a person is asked the standard question, then there is probability 1/2 that she will give a positive response; however if she is asked the sensitive question, then there is an unknown probability p that she will give a positive response. The statistician can observe only the total number X of positive responses that were given by the n persons in the sample. He cannot observe which of these persons were asked the sensitive question or how many persons in the sample were asked the sensitive question. Determine the M.L.E. of p based on the observation X.