Preheat oven to 325 degrees F (165 degrees C). Invert pie shell over another pie pan of equal size. This will keep the crust from shrinking down into the pan. Bake in this position for 10 minutes, until partially baked. Turn right side up, and remove the extra pie pan from inside the crust.
In a large bowl, combine apples, cranberries, and sugar. Cover, and set aside for about 20 minutes. Mix in tapioca, and set aside for 15 to 20 minutes, until tapioca has absorbed fruit juice. Spread mixture into the partially baked pie shell.
In a medium bowl, combine flour, brown sugar, cinnamon, salt, and butter. Work mixture with fingertips until crumbly. Spread mixture over the apple-cranberry filling. Brush exposed pie shell with lightly beaten egg.
Place the pie on a cookie sheet to catch drips. Bake 45 to 60 minutes on the bottom rack of the preheated oven, or until apples are tender when tested with a wooden pick.