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I googled and it showed that the main meaning of “arbitrary” is “based on random choice or personal whim, rather than any reason or system”, while that of “random” is “made, done, happening, or chosen without method or conscious decision.” It seems that they have pretty the same meaning, is there any difference between them?

  • Ask a mathematician. – Hot Licks Sep 26 at 11:40
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The word "conscious" in the definition of "random" gives the clue to the difference. When something happens or is done truly "at random" there is no element of conscious selection. For instance pulling a lottery ticket from an opaque bag is a random selection since there is no tactile clue to let you make a meaningful selection. It's not until the ticket is unfolded that we can tell which number has been selected.

On the other hand an "arbitrary" selection is one made consciously but with no reference to argument or factual consideration. For example instead of drawing counterfoil tickets from a bag someone could choose a number from the range of numbers sold and call out "324" for no good reason. Since the numbers held would have been allocated randomly the result would be similar to drawing the counterfoil to 324 from the bag but the choice would have been arbitrary rather than random.

To look at it another way consider seawater washing over a beach, its path is determined by initial conditions, gravity and the shape of the beach, its path is random. A child digging a channel in the sand, however, chooses where to dig the channel and the water will follow it. However the child has no particular reason for his choice. The path of the water in the channel is arbitrary.

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