Calculate p-value in a two sided Fisher test

Question

I'd wish to know how is calculated p-value in a fisher.test using as alternative hypothesis a two.sided distribution. It looks to me that R firstly find whether the odd ratio is greater or less than 1, then calculate the p-value for the corrisponding case and then multiply it by 2 to obtain the two.sided case.

This is my contingency table:

mytable <- rbind(c(57248,52891),c(51367,50307))

This is the output from a two.sided test:

> fisher.test(mytable,alternative='two.sided')

Fisher's Exact Test for Count Data

data:  mytable
p-value = 2.086e-11
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
 1.042062 1.078306
sample estimates:
odds ratio 
  1.060024 

And here from a greater test:

> fisher.test(mytable,alternative='greater')

    Fisher's Exact Test for Count Data

data:  mytable
p-value = 1.066e-11
alternative hypothesis: true odds ratio is greater than 1
95 percent confidence interval:
 1.044927      Inf
sample estimates:
odds ratio 
  1.060024 

Answer 1

According to R's help file:

Two-sided tests are based on the probabilities of the tables, and take as ‘more extreme’ all tables with probabilities less than or equal to that of the observed table, the p-value being the sum of such probabilities.

Though the two-sided p-values are usually twice the one-sided p-values, it isn't true for smaller counts.

> mytable <- rbind(c(57,52),c(51,50))
> fisher.test(mytable,alternative='two.sided')

Fisher's Exact Test for Count Data

data:  mytable
p-value = 0.8902
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
 0.6032305 1.9145967
sample estimates:
odds ratio 
  1.074292 

> fisher.test(mytable,alternative='greater')

Fisher's Exact Test for Count Data

data:  mytable
p-value = 0.4513
alternative hypothesis: true odds ratio is greater than 1
95 percent confidence interval:
 0.6579587       Inf
sample estimates:
odds ratio 
  1.074292