[fela]

"what are the odds that the results you observed could have arisen by chance?"

If you say it like this it will very easily be misinterpreted. Once your results are in there are two cases: (1) either the null hypothesis is true and you got those results due to chance, or (2) the null hypothesis is false and there was some actual effect outside of the null hypothesis that helped you get the results.

Due to this it is very easy to interpret you statement as referring to the probability of (1).

Two two following definitions of p-values sound similar but are not:

[Correct] The probability of getting the results by chance if the null hypothesis is true P(Results|H0)

[Wrong] The probability that you got the results by chance and thus the null hypothesis was actually true P(H0|Results)

I'm not saying you didn't get it, but somebody reading what you wrote can very easily be fooled. And there are a lot of dead wrong definitions on the web[0][1][2][3].

[0] https://www.americannursetoday.com/the-p-value-what-it-reall...

[1] https://practice.sph.umich.edu/micphp/epicentral/p_value.php

[2] http://natajournals.org/doi/full/10.4085/1062-6050-51.1.04

[3] http://www.cdc.gov/des/consumers/research/understanding_scie...

[RA_Fisher]

The OP is referring to Fisher's p-value rather than the more common Neyman-Pierson method that you refer to. Fisher's method doesn't have the concept of the null hypothesis. The difference is fascinating and I do believe the Fisherian method is superior if you can't easily replicate.

[fela]

The definition of p-value is the same independent of method, as far as I can tell the only real difference is that by Neyman–Pearson you just look at whether the p-value is below a threshold, and Fisher looks at p-value as "strength of evidence" valuable in itself. It's still not the probability that your result was due to chance, it's the probability that under the null hypothesis (and you will definitely need one) you would get that value (or more extreme) by chance.