Three Queen Problem

In the episode entitled 'Maths ' - which can be found in Chapter 3.141592 of the textbook that accompanies the series, the Three Queen Problem is posed to the student.

The Problem
It's the future and Queen Elizabeth III and Queen Elizabeth IV are going to a party held by Queen Elizabeth V. They're keen to make the right impression so it's important that they choose their outfits carefully. Queen Elizabeth III has forty dresses to choose from, where as Queen Elizabeth IV has four thousand. Queen Elizabeth V has just one dress, but it has the ability to transform itself into the shape of any dress. The night before the party Queen Elizabeth IV’s handmaiden steals the paterns to Queen Elizabeth III's dresses and working through till dawn makes forty exact replicas.

Can you calculate the probability that all three queens will be wearing the same dress at the party and how many times can Queen Elizabeth V's dress change before it overheats?

The Solution
After reading through the problem a couple of times it can be readily concluded that the party was cancelled.