I made this puzzle up in 1999 when I was a student.
Fort Alpha, on your border territories, was attacked by The Enemy. You don't know if it fell, or repulsed The Enemy's vicious onslaught. A totally trusted advisor, with a detailed knowledge of the military situation, gave counsel before the attack that the chance of a successful defence was 50%. Unfortunately, there are no rapid communication systems; news of the war relies completely on messengers.
Three of your messengers were given a true report of the battle and set out from Fort Alpha immediately. They travelled separately to forts Beta, Gamma and Delta, which are still under your command. Before collapsing dramatically with exhaustion, these three messengers then relayed the message to three more messengers each, so a total of nine fresh messengers then set out to convey the message to your headquarters.
Now the 12 messengers were hand-picked from the ultra-loyal elite of your troops, so it is very very unlikely, though not impossible, that any given one of them is loyal to The Enemy. Your totally trusted advisor says that the chance of a given messenger being a traitor is 1%. Any messenger who was a traitor would reverse the message relayed to them, and pass on the opposite message, so as to befuddle you into inaction.
None of the messengers spoke to one another except to relay the message. The Enemy's spies, if there are any, maintain total secrecy, so none of the messengers know whether any of the others is a traitor and they each assume each other to be loyal to you. Only traitors would lie.
The three messengers who arrived from Beta, one who arrived from Gamma, and one who arrived from Delta, recount that Fort Alpha fell tragically to The Enemy. The other four claim that it bravely withstood the attack and decimated The Enemy's marauding rabble.
Which story do you believe, and why?