# Conditional on k successess for n independent Bernoulli trials

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?

• The probability of getting any specified string which contains $a$ successes and $b$ fails is $p^a\times q^b$. – lulu Dec 8 '17 at 15:15
• @lulu Is my intuition correct and if so how did you arrive at the above mentioned result p^a×q^b? Do we not need to multiply by (a+b choose a). Also what is the significance of the fact that conditional probability is not dependent on p? – shubham kumar Dec 8 '17 at 17:22
• I couldn't follow your approach. My result follows instantly from the independence of each trial (and the fact that multiplication is commutative). Note that I was computing the ordinary probability, not the conditional probability. To deduce the conditional you must divide by the probability that you get $a$ successes and $b$ failures and for that the binomial coefficient is used. – lulu Dec 8 '17 at 17:48