Ok, so you&#x27;re thinking about a random variable which converges to some value when the null hypothesis is true. This is fine, but it has nothing to do whatsoever with p-values.<p>Let me say that your notation is not very appropriate. It makes no sense to say that P(X|H) converges to 1. If you expect X to converge to C if the null hypothesis is true, you can simply say X-&gt;C. A proper notation involving probabilities would be P(|X-C|&gt;epsilon)-&gt;0 for any positive epsilon (convergence in probability) or maybe P(X-&gt;C)=1 (convergence almost surely).<p>Taking as you suggest X=(#tails&#x2F;#heads), you expect that X-&gt;1 if the coin is fair (I&#x27;m not sure why you find this is not a well-defined null hypothesis, but I don&#x27;t really care). However, P(X)&lt;1 for every X. In fact, P(X=1)-&gt;0 as the number if trials increases (X will get closer to 1 on average, but getting exactly 1 will get more and more unlikely).<p>As I said, you&#x27;re free to prefer your converging statistics and your well-defined null hypothesis. But you should be aware that people are talking about something completely different when discussing things like the 1e-7 p-value in the Higgs boson discovery or the reproducibility of statistically significant results.<p>EDIT: Another example, maybe better-defined: a random variable distributed (under the null hypothesis) x~Normal(mu=0,sigma=1). Let&#x27;s say you take N samples (I let you choose the number, so I don&#x27;t pick one which is not good enough).The statistic is the mean X=(x_1+x_2+..+x_N)&#x2F;N. If the null hypothesis is true, X-&gt;mu=0.  You get X=1&#x2F;sqrt(N). What&#x27;s your &quot;p-value&quot; in that case?