Probability under the null, not of the null
A p-value is the probability of observing data as extreme or more extreme than what you found, assuming the null hypothesis is true. It is not the probability that the null hypothesis is true, nor the probability that a detected effect is real. This distinction is the source of widespread misinterpretation.
A standard normal distribution with shaded critical zones at the tails shows the calculation: if your test statistic lands in the shaded region (p < 0.05), you reject the null at the 5 percent significance level. The shaded area's proportion represents your p-value.
Common misinterpretations and correct meaning
A p-value of 0.03 does not mean there is a 97 percent chance your finding is real. It means if the null were true, you would observe such extreme data 3 percent of the time by random chance. Whether 3 percent is 'extreme enough' to reject the null depends on your prior willingness to accept false positives (the significance level alpha).
Correct interpretation requires understanding the procedure: fix alpha before the experiment, design appropriate sample sizes, and use p-values only as a decision rule for rejecting the null. Mixing p-values with confidence intervals and effect sizes gives a richer picture than p-values alone.