Verified Certificate

Supported by the following organizations

This is to certify that Ali Asghar Karani successfully completed and received a passing grade in XRoad4x: Crossroads: Fat Chance: Probability from the Ground Up a course of study offered by HarvardX, an online learning initiative of Harvard University.

Noted by

Benedict Gross

Benedict Gross

Leverett Professor of Mathematics, Emeritus Harvard University

Joseph Harris

Joseph Harris

Higgins Professor of Mathematics Harvard University

Emily Riehl

Emily Riehl

Assistant Professor, Department of Mathematics Johns Hopkins University

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About edX Verified Certificates

An edX Verified Certificate signifies that the learner has agreed to abide by the edX honor code and completed all of the required tasks of this course under its guidelines, as well as having their photo ID checked to verify their identity.