Question: I have a large dataset from a survey that describes what web pages people use. So for each person I have a list of pages that they visit and how frequently they visit them. What methods can be used to cluster both the webpages (based on similar users who use them) and the users (who use similar webpages).

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Example>
User A visited web pages a, b, a, c
User B visited a, b, d, b
User C visited page e, e

```

Conclusion from that would be that pages `a`,`b`,`c`,`d` are somewhat similar and belong to the same cluster. The other cluster is page `e`. Also, users a,b are similar, the other user cluster is the one containing `User C`, who visits different pages.

Answer: 
You are not correct in the way that in the frequentist framework parameters (in your case probability\proportion) are NOT a random variable. However, if you assume Bayesian framework, you can safely work with parameters as random variables and to obtain probabilities that some parameters exceeds some value.
You have your sampling distribution $$X|p,N \sim B(N,p)$$
Assume that you observed sample is $X\_{1},...,X\_{t}$. Then, the likelihood function is proportional to $$p^{r}(1-p)^{N-r},$$
where $r= \Sigma X\_{k}$. Now you need prior distribution on the proportion parameter. The usual choise in this problem is to use beta distribution $B(\alpha, \beta)$. Then you posterior (due to the conjugacy property) will also be a beta distribution with updated parameters $B(\alpha +r, \beta +N-r)$ or 
$$p|X,N \sim B(\alpha +r, \beta +N-r)$$
Now you can calculate the probabilities that you want, though it is not straightforward, since it requires to calculate incomplete Beta function. Another aspect is that I dont see any way how to calculate the number of experiments, since all statistical information colapses to one sum $r$.
I think Tomas' answer addresses some of the misconceptions you seem to have with your question. First of all the binomial distribution assumes an infinite population where a sample of size N is selected. You do not formulate the problem in a Bayesian framework and yet you do an integration that appears to assume a uniform prior on the proportion.

Tomas shows you how to do the inference in the Bayesian framework using a Beta conjugate prior. In the frequentist framework you can ask what is the maximum likelihood estimate for p given a sample of size n from a binomial distribution where all n observations are failures?

The likelihood function is L(p)=(1-p)$^n$,.

ln[L(p)]= n ln(1-p) d/dp{ln[L(p)]} = -n/(1-p). -ln[l(p)] s maximum at p=0. If p=0 is not an acceptable answer because you know that p>0 and must be at least p$\_0$ then thmaximum likelihood estimate under the constraint would be p=p$\_0$.

In the frequency approach parameters are not given probability distributions. So the question in your title can only be answered using the Bayesian approach. You can integrate the posterior distribution for p in the region p<ϕ to answer the question. The posterior could be the beta specified by Tomas in his answer.