The Not-Quite-So Amazing Case of John O'Donnell

According to the BBC (17th May 2012):

Labour MP John McDonnell has defied odds estimated at 58,000 to 1 to top the annual Private Member's Bill ballot for two years in a row.

That sounds pretty amazing. Where does that estimate come from?

In the next sentence, we learn:

MPs' names are selected at random, with 240 having entered the draw this year.

See what they've done? If we assume (as I suspect the estimator did) that 240 people entered in 2011 as well, then the probabity of Mr. McDonnell's winning in both 2011 and 2012 is indeed 1/(240 x 240) = 1/57,600.

But that's like saying "isn't it amazing that even though the odds on winning the lottery are about 14 million to 1 against, someone wins most weeks?"

I don't know how many people who enter the ballot year to year are the same, but it seems likely it's quite high. Let's assume (conservatively) that it's half. Then the odds of the same person winning in 2011 and 2012 are not 1/57,600 but 1/480.

So it should be a rare event; but not that rare.

And of course, the odds of Mr. O'Donnell's winning in 2012 were 1/240. Just as, if he enters along with 239 others next year, he'll still have a 1/240 chance of winning. That would, however, be more genuinely remarkable.