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You can do Petra by taxi but I don't know how the prices compare...I think all trips in Egypt it is better to do the organised excursions as it is a very long journey to Valley of the Kings (over 2 hours) and you go in a convoy of coaches with security guards. and of course the ship will wait for you if you are delayed!!
Petra is an easy one as you have a two day stop in port and it is easy to arrange either a day trip or an overnight one. However I would NOT recommend individual trips in Egypt as you will have to wait for the convoy anyway so you do not gai.