Tim only wins if two brown eggs are delivered first. Sooner or later, a white egg will follow, e.g. bbw, bbbw, bbbbw, etc. Tim wins in the case of any of these sequences. The probability for two brown eggs to be delivered first is 1/2 * 1/2 = 1/4.
In all other cases, Sarah wins, as the sequence bb is always preceded by a white egg. Sarah therefore has a greater chance (75 %) of winning the bet.