Are you claiming that jury members select a verdict at random? That there is no thought process happening in each of the individuals on the jury? Regardless of whether the juror's beliefs correspond to the actual reality of the case, that individual will still require some reason (evidence) for believing a particular thing. If you are trying to convince a juror of something, you will need to interact with that individual on such a level that you provide them with sufficient evidence for them to believe the claim you want them to believe.
This is separate from the rule of law in the US where a prosecution must prove its allegations beyond a reasonable doubt. It is true that a defense is not required to make any claims or present any evidence. However, depending on the jurors selected, in order to instill or preserve a reasonable doubt, some claims may need to be made. In order for a juror to believe a claim, that individual juror will need to find or receive some evidence sufficient for them to believe the claim.
Out of the millions of trials that have happened I would not be surprised if at least one jury that was undecided simply went with a coin flip or other random means.
Historically this may even have been a unusual, but not that uncommon practice. aka trusting in 'the gods' or some such.
http://en.wikipedia.org/wiki/Testimony