# Can p-value be greater than 1?

Say I perform Student's t-test: can the p-value be greater than 1?

The reason I ask this is that, some say that the p-value is the probability that we will have a Type I error and I would like to show them that p can be greater than 1 so that is not a valid way to think.

• As the answer explains, P-values are probabilities and so cannot exceed 1, so whatever argument you had in mind was fallacious. – Nick Cox Mar 1 '16 at 8:13

The $p$ value, as explained very nicely in this post by @fcop is not the probability of making a type I error, but the probability of getting a value for a test statistic higher than the one we got, under the NULL hypothesis.
We have a fixed type I error decided upon whereby we are ready to accept only a certain risk of excluding $H_o$ incorrectly. But say you set a risk alpha of $0.05$ and the $p$ value obtained ends up being $0.0001$, you will exclude $H_o$ because $0.0001 < 0.05$, as much as if the p value had been $0.04$.
In any event, the p-value is a probability value, and probability measures are bounded between $0$ and $1$.