How To Bind a CheckBox inside a ListBox using MVVM in Windows Phone 8

Bind to the ListBox using the ElementName markup extension then:

<ListBox x:Name="lb" Grid.Row="0" ItemsSource="{Binding ShoppingItems,Mode=TwoWay}" SelectedItem="{Binding SelectedItem,Mode=TwoWay}" FontSize="42" FontWeight="Light" FontFamily="Segoe WP Light" HorizontalContentAlignment="Stretch" ScrollViewer.HorizontalScrollBarVisibility="Visible">
                <i:Interaction.Triggers>
                    <i:EventTrigger EventName="SelectionChanged">
                        <i:InvokeCommandAction Command="{Binding Tapped}"  CommandParameter="{Binding ElementName=List,Path=SelectedItem}"/>
                    </i:EventTrigger>
                </i:Interaction.Triggers>
                <ListBox.ItemTemplate>
                    <DataTemplate>
                        <Grid>
                            <Grid.ColumnDefinitions>
                                <ColumnDefinition Width="Auto"/>
                                <ColumnDefinition Width="*"/>
                            </Grid.ColumnDefinitions>
                            <Grid Grid.Column="0" Margin="0,-19,0,22">
                                <CheckBox Grid.Column="0" IsChecked="{Binding Path=DataContext.IsChecked, ElementName=lb, UpdateSourceTrigger=PropertyChanged, Mode=TwoWay}" />
                            </Grid>
                            <Grid x:Name="MyGrid" Margin="0,-19,0,22" Grid.Column="1">
                                <Grid.ColumnDefinitions>
                                    <ColumnDefinition Width="Auto"/>
                                    <ColumnDefinition Width="Auto"/>
                                    <ColumnDefinition Width="Auto"/>
                                </Grid.ColumnDefinitions>
                                <Grid.RowDefinitions>
                                    <RowDefinition Height="Auto"/>
                                    <RowDefinition Height="Auto"/>
                                    <RowDefinition Height="Auto"/>
                                </Grid.RowDefinitions>
                                <TextBlock Text="{Binding DisplayingItem,Mode=TwoWay}" FontSize="46" Grid.Column="0" Grid.Row="0" TextWrapping="Wrap" />
                                <TextBlock Text="quantity" FontSize="32" Grid.Row="1" Margin="6,0,0,0" Grid.Column="0" TextWrapping="Wrap"/>
                                <TextBlock Text="{Binding DisplayingQuantity,Mode=TwoWay}" FontSize="32" Grid.Column="1" Grid.Row="1" Margin="32,0,12,12" TextWrapping="Wrap"/>
                                <TextBlock Grid.Column="2" Grid.Row="1" FontSize="32" Margin="32,0,12,12" Text="{Binding DisplayingPackaging,Mode=TwoWay}" TextWrapping="Wrap"/>
                                <TextBlock Text="price" Margin="6,0,0,0" FontSize="32" Grid.Row="2" Grid.Column="0" TextWrapping="Wrap"/>
                                <TextBlock Text="$" FontSize="32" Grid.Row="2" Grid.Column="1" Margin="32,0,12,12" TextWrapping="Wrap"/>
                                <TextBlock Grid.Column="3" FontSize="32" Grid.Row="2"  Margin="32,0,12,12" Text="{Binding DisplayingPrice,Mode=TwoWay}" TextWrapping="Wrap"/>
                            </Grid>
                        </Grid>
                    </DataTemplate>
                </ListBox.ItemTemplate>
            </ListBox>