Type II error

A Type II error needs two things at once: $H_0$ is false, and the test fails to reject it. Its probability is $\beta$ (beta), and like power it is defined against one specific alternative value. There is no single $\beta$ for a test, only a $\beta$ for each value the parameter might really have.

Use the z test the power visualizer runs: $H_0: \mu = 100$ against $H_a: \mu > 100$, with $\sigma$ (sigma) known to be 15 and $n = 25$, so the standard error is 3 and the $\alpha = 0.05$ cutoff sits at 104.93. If the truth is $\mu = 105$, the sample mean falls short of that cutoff with probability 0.4913. That is $\beta$, and the matching power is $1 - 0.4913 = 0.5087$. Move the truth out to 110 and $\beta$ drops to 0.0457 with nothing about the test changed.

The sentence to stop writing: "the p-value was 0.31, so there is a 31 percent chance I made a Type II error." A p-value is computed assuming $H_0$ is true, and under a true $H_0$ a Type II error cannot happen at all. $\beta$ is never read off the data. It is a property of the test pointed at an alternative value you do not get to see.

Only one of the two errors is available on any given test, because $H_0$ is either true or false. If $H_0$ is true there is no $\beta$ to speak of and the only mistake on the table is a Type I error. Note also that $\alpha$ and $\beta$ are not complements: here they are 0.05 and 0.4913, which add to nothing meaningful.

Say which lever you are pulling. Holding $\alpha$, the true effect, and the spread fixed, more data shrinks $\beta$: raising $n$ from 25 to 100 takes it from 0.4913 to 0.0457. Raising $\alpha$ shrinks it too, at the price of more false alarms, since $\alpha = 0.10$ puts $\beta$ at 0.3501 in the same test. AP Statistics treats both errors in topic 3.8, Potential Errors When Performing Tests.