NCAA Basketball Tournament :?
The 64-team portion of the tournament involves 63 games (let's exclude the play-in games for now), most of which have an uncertain result [#1 seeds have never lost against a #16 seed since the tournament went to 64 teams in 1985 so that's the closest to a sure thing, but all other match-ups have non-unanimous decisions - even four #15 seeds have won first round games]. If we approximately say that #1 will always beat #16, that leaves 59 games with uncertainty. Even if we assumed #2 always beats #15 & #3 always beats #14 (16 cases otherwise for 3 vs 14), you'd still have 51 games to predict correctly. With 1 billion brackets (~2^30), you could create all possible combinations for 30 of the 51 games you need to predict. That still leaves 21 games you have to answer correctly on your own. Let's assume you save for yourself the 'easiest' games to pick and that each one has an 80% chance of a correct pick. With this, the probability of a perfect bracket is still just under 1%. To your other question about 'rank of a perfect bracket' - I think this depends on what the outcomes were. If you look at 2011, the rank of the perfect bracket would likely be very low on the list of possible outcomes because the Final Four had #3, #4, #8, & #11 seeds. Assuming that the higher seeds are better (i.e. assume that the tournament committee is the all-knowing oracle - skeptics can insert joke here), you can quickly determine that a Final Four prediction with #2, #4, #8, & #11 seeds from the respective regions would be higher ranked, so you'd have at least 2*3*7*10 = 420 brackets which would be higher ranked (the actual number should be a lot more because I've only counted cases where higher seeds in each region made it - if the seeds are roughly equivalent, #1, #1, #7, & #9 would be equally probable as #1, #1, #9 [Old Dominion, #9 below #8 Butler], #7 [UCLA, from #11 VCU's region] but this combination isn't one of the 420. You also have the case where you could have #1, #4, #8, #11 where 3 of the semifinalists are the same but the 4th team is different. That introduces another 2+3+7+10 combinations. I hope this helps some.