Look Around You

Maths can help us in our daily lives - for example, in cooking pasta, in calculating the correct moment for starting a conversation, or (if you can fly) in working out where to land. Queen Elizabeth III and Queen Elizabeth IV are going to a party given by Queen Elizabeth V. QE3 has forty dresses to choose from, QE4 has four thousand, and QE5 has only one dress, but it can take the form of many dresses. Then, during the night, QE3's servant steals the plans for QE4's party dress and, working till dawn, makes four thousand exact copies... So can we reckon what the chance is that all of the Queens will be wearing the same dress at the party?