Question A sequence of n independent experiments is performed. Each experiment is a success with probability p and a failure with probability q = 1 − p. Show that conditional on the number of successes, all valid possibilities for the list of outcomes of the experiment are equally likely.
Attempt Let Xj be 1 if the jth experiment is a success and 0 otherwise, and let X = X1+......+Xn be the total number of successes. Then for any k and any a1,,,,an each of which equal to 1 or 0 a1 +....+ an = k, As far as i can understand, that given the number of success is k, all the possible outcomes with success k are equally likely i.e. 1/(n choose k)
Doubts 1) If this is correct, I am only able to understand this intuitively, however I am not able to obtain a mathematical approach to it. 2) How is the result of this question related to sufficient statistic?