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You are right. But this is actually not a problem if the data is 100% random, because the chance that the message is "Attack from the north" is the same as "Attack from the south", or "I like cats so much!!"
But your key can be very large! You could pick different or random indexes from the key. Maybe that's a solution.
Or come up with something else.
joppiesaus
@Gordon
I guess the problem is how I'd go about changing that key - as I understand it, if I use it to encode multiple messages then it starts to be pretty easy to work out what the key was...
For instance if I just sent 1000 packets and random data over the air using the same pad, you could listen in, take the average of all packets, and you've basically got the key :(