This design is based on my previous larger design that has a larger base.  This design needs much less time to print than the previous one.

The gears were downloaded from McMaster Carr.  The ratio of teeth between the gears is 60:15 (4:1).  So, to turn the thrower gear 90 degrees, the larger drive gear needs to turn 1/4 as much, or 22.5 degrees (if my math is correct).  Since there is a very small load, not much torque is needed to throw the dice as fast as you wish.