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Posted By: Glenn

The specific odds that every team in the NL East would have a winning record and every team in the NL West have a losing record, assuming a 50/50 chance at a winning record for each team, are [10!/(5!5!)]. That is, 252:1.

On the other hand, the odds than all the teams in any division would have a winning record while all the teams in another division in the same league have a losing record (still using the 50/50 assumption) would be substantially higher. I can calculate that if anyone's interested, but the problem is that the records of the different teams are not independent events. If each team has a 50/50 chance of winning each game, the odds that all the teams in the NL West would have losing records are MILLIONS of times higher if all the NL East teams have winning records than if all the NL East teams also had losing records (which is nonetheless possible, as, for example, if no NL Central team had lost an out-of-division game all season, AND all the other NL teams were playing .500 ball against each of their other opponents).

edited to add: The original calculation assumes that half of the 10 teams will have losing records and half will have winning records, and assesses the odds that all 5 of the winning teams are from the East, which is not exactly what Peter asked. Without that assumption, the odds are even easier to calculate. 2^10:1. That is, 1024:1.