# p value and number of predictors

I was going through the An Introduction to Statistical Learning: With Applications in R book and I am facing a bit of difficulty in understanding the below paragraph.I am including the paragraph itself because I do not want to tamper with the original information.

consider an example in which p = 100 and H0 : β1 = β2 =. . . = βp = 0 is true, so no variable is truly associated with the response. In this situation, about 5% of the p-values associated with each variable will be below 0.05 by chance. In other words, we expect to see approximately five small p-values even in the absence of any true association between the predictors and the response. In fact, we are almost guaranteed that we will observe at least one p-value below 0.05 by chance! Hence, if we use the individual t-statistics and associated pvalues in order to decide whether or not there is any association between the variables and the response, there is a very high chance that we will incorrectly conclude that there is a relationship. However, the F-statistic does not suffer from this problem because it adjusts for the number of predictors. Hence, if H0 is true, there is only a 5% chance that the Fstatistic will result in a p-value below 0.05, regardless of the number of predictors or the number of observations.

I am not able to comprehend the relationship between p values and predictors.I tried searching also but everywhere the explanation is regarding p value and sample size only.

The paragraph adresses the problem of multiple testing, and discusses why we cannot just do $$p = 100$$ individual hypothesis tests with $$H_0: \beta_k = 0$$ for each individual predictor at the significance level $$\alpha = 0.05$$. Even if the null hypothesis is true there is still a chance that the specific $$\hat{\beta}_k$$ calculated from your sample leads to a rejection of $$H_0$$. The probability of rejecting $$H_0$$ even if it is true is equal to your significance level. So, in the example from the paragraph with 100 hypothesis tests and a significance level of 5%, the expected number of rejections is $$0.05 \times 100 = 5$$, even if all null hypotheses are true (assuming independence). And, as stated in the paragraph, the probability of observing at least one p-value below 0.05 is close to 1. If the multiple testing is not adjusted for, you will falsely conclude that there is evidence in your data at a 5% significance level of relation between predictor and response.
The paragraph then suggest using the $$F$$-statistic instead, and simultaneously test all $$\beta_k$$'s.