The long "tail" on a detergent molecule is made up mostly of carbon atoms bonded to other carbon atoms. Why would you expect the tail to be no polar?

A detergent molecule consists of a polar ionic head and a non polar hydrocarbon tail. Polarity of a molecule is due to difference in electronegativities of constituting atoms. For example O-H bond is polar because oxygen is very electronegative compared to hydrogen (3.44 vs 2.2) so O acquires a partial negative charge and H acquires a partial positive change making the bond polar.

Detergent tail molecule is non polar because it is made of hydrocarbons i. e. hydrogen and carbon atoms. Carbon atoms are bonded to each other and to hydrogen atoms. Electronegativities of carbon and hydrogen are almost same (2.55 and 2.2 respectively). The difference is not much to induce polarity in C-H bond also hydrocarbon tails have maximum C-C bonds which have no polarity because both the atoms are same. Hence, the tail of detergent molecule is non polar.