Final answer:
The probability P(Y* = y*) is the same as P(n - Y = y*), because the number of failures in a binomial distribution is also binomially distributed, just with the roles of successes and failures inversed.
Step-by-step explanation:
The student is asking about the binomial distribution and the properties of a transformation of a binomial random variable. In this case, the transformation is represented by Y* = n - Y, where Y is the original binomial random variable representing the number of successes in n trials, and p is the probability of success on any given trial. The question is about expressing the probability of a certain number of successes for this transformed random variable.
To show that P(Y* = y*) = P(n - Y = y*), we can argue as follows: Since Y is a binomial random variable, the number of failures is also binomially distributed. Therefore, if Y is the number of successes, then n - Y is the number of failures, which is equivalent to the number of successes if we were counting failures instead. If we define a success now as what we previously considered a failure, the number of 'new successes' will have the same binomial distribution as the original number of successes.
Therefore, the probability of observing y* 'new successes' (or failures in the original sense) is the same as the probability of observing y* failures if we had defined the random variable that way from the beginning, which is by definition the probability of observing n - y successes in the original distribution.