# Conditional expectation when more than certain number of heads are observed out of 'n' trials

Suppose a coin is tossed $$n$$ times and it is then observed that the numbers of head (the success event) appeared is at least $$k. I'm interested in knowing the conditional expectation of no. of heads. If $$h$$ denotes the no. of heads then what is $$E[h|h\geq k]$$? For this, we need to know the conditional pmf of $$h$$. I have two thoughts here:

(i) Conjecturing $$\gamma$$ as the success probability, calculate the conditional pdf. Here I have the following expression: $$\begin{equation} E[h|h\geq k]= \bigg(\frac{1}{p(h\geq k)}\bigg) \big[k.p(h=k)+(k+1).p(h=k+1)+ ....+n.p(h=n)\big] \end{equation}$$ The relevant probabilities involved are nothing but binomial probabilities and are calculated using the conjecture $$\gamma$$. Does this result above have a simpler expression?

(ii) Use MLE method to infer the probability of success first and then use this inferred value to perform (i). Now, I looked hard, I could not find an MLE for this. I know for an observed event $$(h=k)$$, the MLE is $$\hat{\gamma}=k/n$$ but couldn't find the same for the event $$(h \geq k)$$.