Week 12 N.F.L. Game Probabilities, and a Tired Theory The Giants' offense peaked in Week 5 against Cleveland and has been in a relatively consistent decline since. The Giants' worst outing was their most recent one, against Cincinnati. This has spurred theories of a tired arm for quarterback Eli Manning because of an accumulation of instances with a high number of pass attempts, like 51 against Tampa Bay, 42 against Philadelphia, 40 against Washington and 46 against the Bengals. But this is the worst kind of post-hoc analysis. Manning has attempted just over 36 passes per game this season, which is no more than what he averaged in the regular season last year, and the 10th-most attempts for a quarterback this season. In last season's Super Bowl-winning campaign, Manning had 742 attempts, about twice as many as in 2012 so far. With 364, Manning has thrown as many times as a major league baseball starting pitcher after four starts. Quarterbacks make most of their throws during the week in practice, with only a small portion during a game. The tired arm hypothesis is particularly suspicious when you consider our neighbors to the north. The Canadian Football League is even more pass happy than the N.F.L., with a run-pass ratio of 35 to 65, mostly because there is one fewer down. The C.F.L. also has an 18-game regular schedule, and there have been no reports of arms falling off. I think the tired arm theory stems from an inverse correlation between success and pass attempts. The better that teams play, the fewer attempts they need to make. Teams with leads take fewer risks and keep the clock ticking by running the ball. It's a well known and understandable paradox, but the illusion that the extra passing causes losing continues to be seductive. Had someone said that Manning (or any other quarterback or set of quarterbacks) was passing more frequently and made the prediction that they would be less successful, the theory would have more merit. But looking backward at the coincidence of a few extra pass attempts over a few weeks and declining performance is not good evidence of causation. And to the Giants' and Manning's credit, they didn't entertain the theory themselves. Manning brings his arm into an N.F.C. showdown with the Packers on Sunday night. If he plays well, you can bet some analysts will attribute it to the week of rest from the Giants' recent bye week. Here are this week's game probabilities, tired and rested arms in all.