Binding "SelectedItems" of a Listbox

Question

Could you please help me figure out how I could bind "SelectedItems" of a listbox?

Here I have a 'listbox_2' that contains a list of fruits and vegetables ;) There's another 'listbox_1' that contains groups of fruits and vegetables (under a list format...maybe is it my first error? Should it rather be a list of fruits/vegetables?) and a recipe.

I would like that on each group selection (in listbox_1), the corresponding fruits and vegetables would be selected (in listbox_2)...and make it possible to modify that list

Unfortunately, the way to achieve this behavior stays obscure to me...could you please help?

Here's what I've set up untill now:

C#

public partial class MainWindow : Window
{
    ItemList il;
    GroupList gl;

    public MainWindow()
    {
        InitializeComponent();
    }
    private void Window_Loaded(object sender, RoutedEventArgs e)
    {
        il = new ItemList();
        ICollectionView cvs = CollectionViewSource.GetDefaultView(il);
        cvs.SortDescriptions.Add(new SortDescription("_type", ListSortDirection.Ascending));
        cvs.SortDescriptions.Add(new SortDescription("_name", ListSortDirection.Ascending));
        cvs.GroupDescriptions.Add(new PropertyGroupDescription("_type"));
        ListBox2.ItemsSource = cvs;

        gl = new GroupList();
        ICollectionView cvt = CollectionViewSource.GetDefaultView(gl);
        ListBox1.ItemsSource = cvt;
    } 
}

public class Item
{
    public string _type { get; set; }
    public string _name { get; set; }
    public Item()
    {
    }
}
public class ItemList : ObservableCollection<Item> {
    public ItemList() {
        base.Add(new Item() { _type = "fruit", _name = "apple" });
        base.Add(new Item() { _type = "vegetable", _name = "potato" });
        base.Add(new Item() { _type = "fruit", _name = "banana" });
        base.Add(new Item() { _type = "vegetable", _name = "tomato" });
        base.Add(new Item() { _type = "fruit", _name = "pear" });
        base.Add(new Item() { _type = "vegetable", _name = "salad" });
        base.Add(new Item() { _type = "fruit", _name = "orange" });
        base.Add(new Item() { _type = "vegetable", _name = "onion" }); 
    }
}

public class Group
{
    public string _groupname { get; set; }
    public List<String> _members { get; set; }
    public string _recipe { get; set; }
    public Group()
    {
    }
}
public class GroupList : ObservableCollection<Group>
{
    public GroupList()
    {
        base.Add(new Group() { _groupname = "Group_1", _members = new List<String>() { "apple", "salad" }, _recipe = "Do this and do that" });
        base.Add(new Group() { _groupname = "Group_2", _members = new List<String>() { "banana", "onion" }, _recipe = "Don't do that and do this" });
    }
}

XAML

<Window x:Class="WpfApplication1.MainWindow"
    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
    Title="MainWindow" Height="350" Width="525"
    Loaded="Window_Loaded">
<Grid>
    <Label Margin="12,0,378,283" Content="Group"></Label>
    <Label Margin="190,0,200,283" Content="Members"></Label>
    <Label Margin="309,0,81,283" Content="Recipe"></Label>
    <TextBox Margin="309,34,12,12" DataContext="{Binding SelectedItem, ElementName=ListBox1}" Text="{Binding Path=_recipe, Mode=TwoWay, UpdateSourceTrigger=PropertyChanged}"></TextBox>
    <ListBox Margin="12,34,378,12" Name="ListBox1" SelectionMode="Single">
        <ListBox.ItemTemplate>
            <DataTemplate>
                <TextBlock Text="{Binding _groupname}" />
            </DataTemplate>
        </ListBox.ItemTemplate>
    </ListBox>
    <ListBox Margin="190,34,199,12" Name="ListBox2" SelectionMode="Multiple" SelectedValuePath="_name">
        <ListBox.ItemTemplate>
            <DataTemplate>
                <TextBlock Text="{Binding _name}" />
            </DataTemplate>
        </ListBox.ItemTemplate>
        <ListBox.GroupStyle>
            <GroupStyle>
                <GroupStyle.ContainerStyle>
                    <Style TargetType="{x:Type GroupItem}">
                        <Setter Property="Template">
                            <Setter.Value>
                                <ControlTemplate>
                                    <Expander Header="{Binding Name}" IsExpanded="True">
                                        <ItemsPresenter />
                                    </Expander>
                                </ControlTemplate>
                            </Setter.Value>
                        </Setter>
                    </Style>
                </GroupStyle.ContainerStyle>
            </GroupStyle>
        </ListBox.GroupStyle>
    </ListBox>
</Grid>

EDIT: from my reading, it seams that listboxes' 'SelectedItems' property is read-only..maybe then a solution to bind to a checkbox component inside the listbox item?

Thanks for any kind help!