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Electrical engineering (EE) is one of the largest engineering disciplines. Computer engineering (CpE) has a similar curriculum, but different demographics and student outcomes. Using a dataset from universities in the U.S. that includes over 70\,000 students who majored in engineering, this paper describes the outcomes for students matriculating in and migrating into EE and CpE so as to inform the decision making of faculty, department heads, and deans. Although men consistently outnumber women in EE and CpE, the rates of matriculation and six-year graduation vary by race and gender. EE is the most popular choice for Asian and Black students (males and females) at matriculation, but while Asians graduate at high rates, Blacks (particularly males) are not retained. Retention is higher in EE than in CpE despite the similarity of the curricula. Graduation rates are lower than expected for women of all races in CpE and for Hispanic women in EE. By the third semester, some students of all races and genders have left their matriculation major, but others have migrated in from other majors, compensating for some of this loss. CpE students are more likely than EE students to switch to another major. Trajectories of EE and CpE students are racialized and distinct. CpE loses more students and attracts fewer students than EE. These findings illustrate the importance of disaggregating by engineering major as well as race and gender to improve recruitment and retention overall. [ABSTRACT FROM AUTHOR]