<DataGrid Name="SearchResultTable" AutoGenerateColumns="False" MinHeight="200" CanUserSortColumns="False">
    <DataGrid.Columns>
        <DataGridTextColumn Header="lastName" Binding="{Binding LastName}" SortDirection="Ascending" IsReadOnly="True" DisplayIndex="0"/>
        <DataGridTextColumn Header="firstName" Binding="{Binding FirstName}"   IsReadOnly="True" DisplayIndex="1"/>
        <DataGridTextColumn Header="dateOfBirth" Binding="{Binding DateOfBirth }"   IsReadOnly="True" DisplayIndex="2"/>
    </DataGrid.Columns>
</DataGrid>

DataGridTextColumn col = (DataGridTextColumn)SearchResultTable.Columns[2];
col.Binding.StringFormat = CultureInfo.CurrentUICulture.DateTimeFormat.ShortDatePattern;

IList<Person> searchResult = // read data from database
SearchResultTable.ItemsSource = searchResult;

Binding cannot be changed after it has been used