Probability that a random person is both X and Y I'm wondering how the following probability problem should be approached: Which probability is the smallest, given a random person? a) The person is a student b) The person is bilingual c) The person is a student and bilingual d) Insufficient information The reason I think (c) may be the answer: $Pr(student, bilingual) = Pr(student | bilingual)Pr(bilingual) = Pr(bilingual | student)Pr(student)$ So it seems to me that $Pr(student,bilingual)$ should be smaller than the probability of one of them. But, and may I'm overthinking this, but I'm starting to have second thoughts as to whether the intended answer might be (d) instead. What if we assume $Pr(student) = 0$ for example? Then $Pr(student,bilingual) = 0$ as well. Is this an unjustified assumption? I'd appreciate any help. Thanks.

$P(\text{student}, \text{bilingual}) \le P(\text{student})$ and $P(\text{student}, \text{bilingual}) \le P(\text{bilingual})$.

This is a consequence of the fact that $A\cap B\subseteq A$ and $A\cap B\subseteq B$ and for $A\subseteq B$ one has $P(A)\le P(B)$.

Depending on your assumption on the sampling population, if you have at least a student who is not bilingual and and at least a bilingual who is not student, then the strict inequalities

$P(\text{student}, \text{bilingual}) < P(\text{student})$ and

$P(\text{student}, \text{bilingual}) < P(\text{bilingual})$ also hold.

Since your problem does not imply any restriction on the sample space, I am inclined to say c). Which means that if you just go out and ask somebody on the street, you will find bilinguals who are not students and students who are not bilinguals.