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Doctors throw antibiotics at viral infections as a placebo cureThat's actually not what doctors say. When they can't determine if an infection is viral or bacterial, they treat it on the chance that it's bacterial (since we have a known treatment for bacterial infections and don't usually have a treatment for viral infections).
Tonsillitis is a good example: About 50% of tonsillitis cases are viral and 50% bacterial. "Relying on clinical presentation and history alone is unreliable in differentiating streptococcal from viral tonsillitis."[1] And bacterial tonsillitis can be quite serious.
So it seems reasonable to go ahead and treat with antibiotics even when there's a 50% chance that it's wasted, because the other 50% chance was that the treatment was highly beneficial.
[1] http://bestbets.org/bets/bet.php?id=2024