Why use Bonferroni approximation for experiment-wise alpha? It seems the Bonferroni method (dividing experimentwise alpha by # of comparisons) for choosing the p level to fix the experimentwise alpha (when doing many pairwise comparisons) is more conservative than just solving $1 - (1 - p)^k = .05$ to get the alpha to use for each of the $k$ pairwise comparisons.  Why not just solve the equation?
 A: The Dunn-Sidak approach (your second equation) is only valid if your contrasts are orthogonal, whereas the Bonferroni adjustment is guaranteed to always hold your experimentwise alpha below your chosen value.  In addition, with only a few contrasts, both methods will yield almost identical values.  When combined with the fact that the Bonferroni adjustment is easier for many people to understand, remember, and calculate, those are the basic reasons for its prevalence.  
A: Gung gave an excellent answer.  i just want to add that nothing is simpler than dividing alpha by n (the number of tests).  Although Bonferroni is conservative it is easy to use and does not require added assumptions as Sidak and others do.  Now it is bad when individual p-values are realtively large or n gets very large.  But when the raw p-values are small it works well adn the level of conservatism is minimal.
A: According to Abdi (2007), the main reason is simply that Bonferroni was easier to calculate than Sidak in the old days.
What the article may disagree with gung's answer is that similar to Bonferroni, when the comparisons are dependent, Sidak also gives a lower bound.  Bonferroni is simply the first (linear) term of Taylor expansion of Sidak. Therefore, you can always use Sidak whether the comparisons are independent or not.
In modern days, Bonferroni is only preferred over Sidak because it is more familiar.  


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*Abdi, H. (2007). The Bonferonni and Šidák Corrections for Multiple Comparisons. Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.

