C# – listbox isSelected databinding in DataTemplate

I try to simply databind IsSelected property with IsSelected field in my class. But after I change the value in code, it doesn't change the property, neither does clicking on ListBoxItem change the field value.

XAML:

<FlipView ItemsSource="{Binding Source={StaticResource itemsViewSource}}" ... >
    <FlipView.ItemTemplate>
        <DataTemplate>
            <UserControl Loaded="StartLayoutUpdates" 
                Unloaded="StopLayoutUpdates">
                <!-- other controls -->
                <ListBox Grid.Row="1" Grid.ColumnSpan="3"
                    SelectionMode="Multiple" VerticalAlignment="Center" 
                    ItemsSource="{Binding Answers}">
                    <ListBox.Resources>
                        <local:LogicToText x:Key="logToText" />
                    </ListBox.Resources>

                     <!-- bind IsSelected only in one way from 
                         code to content --> 
                     <ItemsControl.ItemTemplate>
                        <DataTemplate>
                            <ListBoxItem 
                              IsSelected="{Binding IsSelected, Mode=TwoWay, Converter={StaticResource logToText}}" 
                              Content="{Binding IsSelected, Mode=TwoWay, Converter={StaticResource logToText}}">

                            </ListBoxItem>

                        </DataTemplate>
                    </ItemsControl.ItemTemplate>


                    <!-- not working at all
                    <ListBox.Resources>
                        <Style TargetType="ListBoxItem">
                            <Setter Property="IsSelected" 
                                Value="{Binding IsSelected, Mode=TwoWay}"/>
                            <Setter Property="Content" 
                                Value="{Binding IsSelected, Mode=TwoWay}"/>
                        </Style>
                    </ListBox.Resources>-->

                </ListBox>
            </UserControl>
        </DataTemplate>
    </FlipView.ItemTemplate>
</FlipView>

Code:

Answers

private ObservableCollection<PrawoJazdyDataAnswer> _answers = 
    new ObservableCollection<PrawoJazdyDataAnswer>();
public ObservableCollection<PrawoJazdyDataAnswer> Answers 
{ 
    get 
    { 
       return this._answers; 
    }  
}    

Single item(Answer)

public class PrawoJazdyDataAnswer : NPCHelper// PrawoJazdy.Common.BindableBase
{
    public PrawoJazdyDataAnswer(String ans, bool ansb)
    {
        this._ans = ans;
        this._isSelected = ansb;
    }

    public override string ToString() 
    { 
        return _isSelected.ToString();  //Only For debug purposes 
                                        //normally return _ans 
    }
    private string _ans;
    public string Ans
    {
        get { return this._ans; }
        //set { this.SetProperty(ref this._ans, value); }
    }

    private bool _isSelected;
    public bool IsSelected
    {
        get { return this._isSelected; }
        set
        {
            _isSelected = value;
            FirePropertyChanged("IsSelected");
            //this.SetProperty(ref this._isSelected, value); 
        }
    }
}

FirePropertyChanged

public class NPCHelper : INotifyPropertyChanged
{
    public event PropertyChangedEventHandler PropertyChanged;
    public void FirePropertyChanged(string prop)
    {
        if (PropertyChanged != null)
            PropertyChanged(this, new PropertyChangedEventArgs(prop));
    }
}

Converter(which sometimes seems to be needed and others not…, I tried ~10 approaches from different tutorials/examples)

public class LogicToText : IValueConverter
{
    /// <summary>
    /// 
    /// </summary>
    public object Convert(object value, Type targetType, 
                          object parameter, string language)
    {
        //if (value == null || (bool)value == false)
          //  return "False";

        return value.ToString();
    }

    /// <summary>
    /// 
    /// </summary>
    public object ConvertBack(object value, Type targetType, 
                              object parameter, string language)
    {
        return value.ToString().Contains("True") ? true : false;
    }

Thanks in advance, and sorry for my English(still learning).

@edit
Thanks for quick reply.

For test purposes I created a button and text block:

<Button Click="spr" >Sprawdź</Button>
<TextBlock Text="{Binding Answers[0].IsSelected, Mode=TwoWay}" > </TextBlock> 

It's in other controls part (above list box, but in FlipView)
Click method

private void spr(object sender, RoutedEventArgs e)
    {
        var ans = ((PrawoJazdyDataQuestion)this.flipView.SelectedItem).Answers;
        foreach (var item in ans)
            item.IsSelected = item.IsSelected ? false : true;
    }

As I wrote, when i'm changing data from code side, it is changing content of element, but not appearance of ListBoxItem. And if I just select it on ListBox, it doesn't change the data in TextBlock neither in ListBox itself.

@edit2 fixing typos…