This paper uses a twin construct to test how the thickness of the veil of ignorance (VOI) affects the perception of fairness and redistributional choices. A fortune reversal is generally perceived to be fair behind a thick VOI, but deemed unfair behind a transparent VOI, particularly if one is currently in a stronger social position. A significant association exists between the perception of fairness and the certainty levels about social position. A 50-50 split is preferred with thicker VOI; whilst the status quo is favoured with thinner VOI. Regardless of the VOI's thickness, few opt for a winnertake-all redistribution.