Many discussions on serendipitous research discovery stress its unfortunate immeasurability. This unobservability may be due to paradoxes that arise out of the usual conceptualizations of serendipity, such as “accidental” versus “goal-oriented” discovery, or “useful” versus “useless” finds. Departing from a different distinction drawn from information theory—bibliometric redundancy and bibliometric variety—this paper argues otherwise: Serendipity is measurable, namely with the help of altmetrics, but only if the condition of highest bibliometric variety, or randomness, obtains. Randomness means that the publication is recommended without any biases of citation counts, journal impact, publication year, author reputation, semantic proximity, etc. Thus, serendipity must be at play in a measurable way if a paper is recommended randomly, and if users react to that recommendation (observable via altmetrics). A possible design for a serendipity-measuring device would be a Twitter bot that regularly recommends a random scientific publication from a huge corpus to capture the user interactions via altmetrics. Other than its implications for the concept of serendipity, this paper also contributes to a better understanding of altmetrics’ use cases: not only do altmetrics serve the measurement of impact, the facilitation of impact, and the facilitation of serendipity, but also the measurement of serendipity.