Equation 13 · The Benchmarks That Don't Need the Image
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
Read it piece by piece
Symbol P_chance
hance is part of the quantity the equation computes from the expression on the right.
Symbol k
k occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Symbol N
N is an input to the expression that computes the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
superscript
A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.
See an illustrated explanation →How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
It is worth seeing precisely why an all-or-nothing rule across permutations is so stringent, and where the intuition needs a caveat. If a model’s answer on each rotated pass were an independent uniform guess among k options, the probability of passing all N rotations by chance would be . which for a four-option item run through all four rotations is (1/4)^4 , under half a percent. That is not a real prediction of model behavior — actual models are not memoryless guessers, and a model with a fixed positional bias will do considerably worse than this floor once the correct answer rotates away from its favored slot, while a model with genuine content understanding will pass…
Read the full surrounding passage
It is worth seeing precisely why an all-or-nothing rule across permutations is so stringent, and where the intuition needs a caveat. If a model’s answer on each rotated pass were an independent uniform guess among k options, the probability of passing all N rotations by chance would be . which for a four-option item run through all four rotations is (1/4)^4 , under half a percent. That is not a real prediction of model behavior — actual models are not memoryless guessers, and a model with a fixed positional bias will do considerably worse than this floor once the correct answer rotates away from its favored slot, while a model with genuine content understanding will pass reliably regardless of k or N . The independence assumption is a deliberate simplification, useful only for showing why the two behaviors CircularEval separates — answering from content, and answering from position — land so far apart in practice, exactly as the OpenFlamingo v2 result illustrates.
Sources cited in the article section
These citations give research context. Read each source to check which claims it supports.
Return to The Benchmarks That Don't Need the Image