# Simple probability

At a dance there are $n=3$ married couples: Ann and Andy, Betty and Boris, and Danielle and Dan. The wives select a husband at random with whom to share a dance. What is the probability that each of the three men dances with a woman other than his spouse.

In each pair, I denoted wives with a lowercase letter and husbands with an uppercase. The pairs are aA, bB, and dD. I counted $9$ ways in which women could select men. aA, aB, aD bB, bA, bD dD, dA, dB

So there are $6$ ways to select other spouses. The probability is $\frac{6}{9}$, which is $0.67$, but the answer is $0.33$. What's wrong? Maybe I misunderstood the question, because English is my second language. Please help.

• Which triples of assignments are simultaneously compatible? For instance $\{aA, aB, \langle\text{anything}\rangle\}$ is not a possibility. The question is not about choices taken independently, it's about simultaneous triples of choices that are compatible. (An example of a compatible triple is $\{aB, bD, dA\}$. What fraction of compatible triples have no wife-husband pairing at all?) Mar 9, 2014 at 2:30