Who first used/invented p-values? I am attempting to write a series of blog posts on p-values and I thought it would be interesting to go back to where it all started - which appears to be Pearson's 1900 paper. If you are familiar with that paper, you'll remember that this covers goodness-of-fit testing.
Pearson is a bit loose with his language when it comes to p-values. He repeatedly uses the "odds" when describing how to interpret his p-value. For example, on p.168 when talking about the results of repeat rolls of 12 dice, he says "...which leads us to P=.0000016, or the odds are 62,499 to 1 against such a system of deviation on a random selection. With such odds it would be reasonable to conclude that the dice exhibit bias towards the higher points."
In this article, he refers to earlier work, including an 1891 book on least squares by Merriman.
But Pearson does lay out the calculus for p-values (w.r.t. chi-squared goodness of fit testing).
Was Pearson the first person to conceive of p-values? When I do a search on p-values, Fisher is mentioned - and his work was in the 1920s.
Edited: and a thank you for the mention of Laplace - he did not seem to address the null hypothesis  (Pearson appears to do so implicitly, although he never used that term in his 1900 paper). Pearson looked at goodness of fit testing from: assuming the counts are derived from an unbiased process, what is the probability that the observed counts (and counts more deviant) arise from the assumed distribution?
His treatment of the probabilities/odds (he converts the probabilities to odds) suggests he is working with an implicit idea of the null hypothesis. Crucially, he also mentions that the probability arising from the x^2 value shows the odds "against a system of deviations as improbable or more improbable than this one" - language we recognise now - with respect to his calculated p-values.
Did Arbuthnot go that far?
Feel free to put your comments in as answers. It would be nice to see a discussion.
 A: Jacob Bernoulli (~1700) - John Arbuthnot (1710) - Nicolaus Bernoulli (1710s) - Abraham de Moivre (1718)
The case of Arbuthnot1 see explanation in note below, can also be read about in de Moivre's Doctrine of Chance (1718) from page 251-254 who extends this line of thinking further.
De Moivre makes two steps/advancements:

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*The normal approximation of a Bernoulli distribution, which helps to easily calculate probabilities for results being within or out a certain range. In the section before the example about Arbuthnot's case, de Moivre writes about his approximation (now called the Gaussian/normal distribution) for the Bernoulli distribution. This approximation allows to easily calculate a p-value (which Arbuthnot could not do).


*Generalization of Arbuthnot's argument. He mentions that "this method of reasoning may also be usefully applied in some other very interesting inquiries". (which may give partial credit to de Moivre for seeing the general applicability of the argument)


*

*According to de Moivre, Jacob Bernoulli wrote about this problem in his Ars Conjectandi. De Moivre names this in English 'Assigning the limits within which, by the repetition of experiments, the probability of an event may approach indefinitely to a probability given', but the original text by Bernouilli is in Latin. I do not know sufficient Latin to be able to figure out if Bernoulli was writing about a concept like the p-value or more like the law of large numbers. Interesting to note is that Bernouilli claims to have had these ideas for 20 years (and also the work 1713 was published after his death 1705 so it seems to precede the date 1710 mentioned in the comments by @Glen_b for Arbuthnot).


*One source of inspiration for de Moivre was Nicolaus Bernouilli, who in 1712/1713 made the calculations for the probability of the number of boys being born is not less than 7037 and not bigger than 7363, when 14000 is the number of total born kids and the probability for a boy is 18/35.
(The numbers for this problem were based on 80 years of statistics for London. He wrote about this in letters to Pierre Raymond de Montmort published in the second edition (1713) of Montmort's Essay d'analyse sur les jeux de hazard.)
The calculations, which I did not quite follow, turned out a probability of 43.58 to 1. (Using a computer summing all terms probability of a binomial from 7037 up to 7363 I get 175:1 so I may have misinterpreted his work/calculation.)

1: John Arbuthnot wrote about this case in An argument for divine providence, taken from the constant regularity observed in the births of both sexes (1710).
Explanation of Arbuthnot's argument: the boy:girl birth ratio is remarkably different from the middle. He does not calculate exactly the p-value (which is not his goal), but uses the probability to get boys>girls 82 times in a row $$\frac{1}{2}^{82} \sim \frac{1}{4 \,8360\,0000\,0000\,0000\,0000\,0000}$$ arguing that this number would be even more small when you would consider that one could take a smaller range and that it happened more than in just London and 82 years, he ends up at the conclusion that it is very unlikely and that this must be some (divine) providence to counter the greater mortality among men to finally end up with equal men and women.


 Arbuthnot: then A’s Chance will be near an infinitely small Quantity, at least less than any assignable Fraction. From whence it follows that it is Art, not Chance that governs.

