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Consider an actor making selection decisions using a series of classifiers, which we term a sequential screening process. The early stages filter out some applicants, and in the final stage an expensive but accurate test is applied to the individuals that make it to the final stage. Since the final stage is expensive, if there are multiple groups with different fractions of positives at the penultimate stage (even if a slight gap), then the firm may naturally only choose to the apply the final (interview) stage solely to the highest precision group which would be clearly unfair to the other groups. Even if the firm is required to interview all of those who pass the final round, the tests themselves could have the property that qualified individuals from some groups pass more easily than qualified individuals from others. Thus, we consider requiring Equality of Opportunity (qualified individuals from each each group have the same chance of reaching the final stage and being interviewed). We then examine the goal of maximizing quantities of interest to the decision maker subject to this constraint, via modification of the probabilities of promotion through the screening process at each stage based on performance at the previous stage. We exhibit algorithms for satisfying Equal Opportunity over the selection process and maximizing precision (the fraction of interview that yield qualified candidates) as well as linear combinations of precision and recall (recall determines the number of applicants needed per hire) at the end of the final stage. We also present examples showing that the solution space is non-convex, which motivate our exact and (FPTAS) approximation algorithms for maximizing the linear combination of precision and recall. Finally, we discuss the `price of' adding additional restrictions, such as not allowing the decision maker to use group membership in its decision process.