You didn't mention what your experience level is and what you've already tried.
Part of the key about creating projects like these is breaking it down into smaller steps, and dealing with the smaller step individually. While you have at least started the process of breaking it down and thought about how to do some of the pieces, of using a pressure sensor, etc. you need to break it down further. The questions below might be snarky, it is more to try and get you to see how to break down projects into several sub-steps, rather than sitting back and hoping somebody will write the whole code for you, and give it to you free of charge.
One step would be read from a pressure sensor and if it is true, turn on the on board LED connected to digital output 13 on the Uno. Have you programmed the Arduino to do this and run it with the real sensor? Note, for the final project since you will be having people stand on the plate, have you figured out a sensor that is sensitive enough for a kid to trigger it, but will continue to work if several overweight people step on the platform?
Another step would seem to be building the platform, or at least putting the sensor under a rug and taping the wires which has nothing to do with an Arduino, but it is still crucial to the project.
Also, have you physically built the platform to hold the video screen and speaker so that people standing on the switch can see and hear the video? Have you adjusted the angle of view of the monitor and adjusted the sound so it can be heard, but doesn't blast the people?
Next up would seem to be triggering the video. Have you hooked up an Arduino so that when a switch is pressed, it starts a single video? By starting the video, I mean actually starting the video to make sure you can start the video from the Arduino.
Have you dealt with timing issues, so once the video is triggered, you don't read from the button until after the video is finished?
Have you kept track of the button state in a global/static variable so that even after the timeout, you don't start the video again until the person gets off the mat, and the next person steps on?
Have you set it up to have two buttons, and when you press the first button, it triggers the first video, and when you trigger the second button it triggers the second video? This is to make sure that you can start either video depending on the programming of the Arduino. And here, I mean actually having the second video unit attached? Rather than two buttons, you could use a global/static state variable, and the first button press do the first video, and the second button press do the second video.
Finally, we get to what on the surface seems to be your question of how to trigger a video at random, but is actually the simple part. In things like this, using your favorite search engine can often times give you the answer before asking the question on a general forum. http://lmgtfy.com/?q=arduino+random
Does your project need to run on batteries or can you use a wall socket? Assuming a wall socket, do you have appropriate length of electrical cord for the Arduino and video? Are the cords out of sight, or have you taped them down so that people won't trip over them? If it is batteries, have you figured out how much current draw there is and what batteries to use?
Is the project going to be self contained without somebody near by? Do you have to worry about physical security to prevent people from stealing your Arduino and/or the video setup?