A test for a disease gives an answer "Positive" or "Negative". Because of experimental error, the test gives an incorrect answer with probability 1/4. That is, if the person has the disease, the outcome is negative with probability 1/4 and if the person does not have the disease, the outcome is positive with probability 1/4. You can assume that the probability an error is made in one test is independent of the outcomes of any other tests. In order to improve the probability of getting the correct answer, the test is run n times and the majority outcome is used as the final answer. If n = 3, what is the probability of getting the wrong answer? What about n = 5?

Introduction: Denote X as the random variable of interest here, which is the number of wrong test results out of n runs of the test. Consider getting a wrong test result as a success. It is given that the results are independent of each other, with probability of “success” or probability of getting a wrong test result being p = ¼ or 0.25. Hence, it may be assumed that, X has a binomial distribution with parameters, n, p = 0.25. Since p denotes the probability of success, the probability of failure is, q = 1 – p = 0.75.Calculation: If X ~ Binomial (n, p), then the probability mass function of X is: Here, n is the number of independent trials, p is the probability of success in each trial, and q is the probability of failure. For n = 3, the probability mass function of X is: It is mentioned that the majority outcome in the n tests would be considered as the final answer. In order for the final answer to be wrong, majority of the test results should be wrong. For n = 3, if X = 0 or 1 test result is wrong, then 3 or 2 (respectively) test results would be correct. In such a situation, the final answer would still be correct. This is not the situation we are interested in. Thus, in order for the final answer to be wrong, at least 2 test results should be wrong, that is, the final answer would be wrong if the number of wrong test results is X = 2 or 3, or X ≥ 2. The corresponding probability is calculated below: Thus, the probability of getting a wrong answer when n = 3 is 0.15625. For n = 5, the probability mass function of X is: For n = 5, if X = 3 or 4 or 5 test results are wrong, then 2 or 1 or 0 (respectively) test results would be correct. In such a situation, the final answer would be wrong. Thus, in order for the final answer to be wrong, at least 3 test results should be wrong, that is, the final answer would be wrong if the number of wrong test results is X = 3 or 4 or 5, or X ≥ 3. The corresponding probability is calculated below: Thus, the probability of getting a wrong answer when n = 5 is 0.103515625.