  A new lower bound for the average probability or error for a two-user
discrete memoryless (DM) multiple-access channel (MAC) is derived. This bound
has a structure very similar to the well-known sphere packing packing bound
derived by Haroutunian. However, since explicitly imposes independence of the
users' input distributions (conditioned on the time-sharing auxiliary variable)
results in a tighter sphere-packing exponent in comparison to Haroutunian's.
Also, the relationship between average and maximal error probabilities is
studied. Finally, by using a known sphere packing bound on the maximal
probability of error, a lower bound on the average error probability is
derived.
