The arrangement of phosphate groups facing towards the outside and fatty acids facing inward in the phospholipid bilayer of the plasma membrane is due to the unique properties of these molecules. Phosphate groups contain a negatively charged oxygen atom, making them hydrophilic or attracted to water molecules. Therefore, they naturally orient themselves towards the watery extracellular environment and the cytoplasmic region of the cell.
On the other hand, fatty acids consist of long hydrocarbon chains, which are nonpolar and hydrophobic or repelled by water. As a result, they tend to cluster together away from water. This leads to the formation of a lipid bilayer, where the hydrophilic phosphate groups face the water-based environments (outside and inside of the cell), while the hydrophobic fatty acids remain sandwiched in the middle, forming a barrier that prevents water and water-soluble molecules from freely passing through the membrane.
In summary, the arrangement of phosphate groups facing outward and fatty acids facing inward is a result of the amphipathic nature of phospholipids, with their distinct hydrophilic and hydrophobic regions, which allows for the formation of the phospholipid bilayer structure in the plasma membrane.