A note on the probability of a groupoid having deficient sets

A subset $X$ of a groupoid is said to be deficient if $\mathrm{\mid }X\cdot X\mathrm{\mid }\le \mathrm{\mid }X\mathrm{\mid }$. It is well-known that the probability that a random groupoid has a deficient $t$-element set with $t\ge 3$ is zero. However, as conjectured in [4], we show that the probability is not zero in the case of sets of two elements and calculate the exact value. We explore some generalisations on deficient sets and their likelihoods.