[Siegmund]

An exact calculation would be very difficult.

This looked like a fun problem to spend an hour simulating this afternoon, though, so I did.

The results will vary slightly with what range of hands you play. For my simulation I used the following:

10-handed table;
First 3 seats: play 66,T9s,QTs,KTs,A8s,ATo,KQo, or better
Middle 3 seats: play 22,98s,QTs,KTs,A5s,ATo,KJo, or better
CO/Button: 22,87s,QTs,K9s,Axs,Axo,KT,QT,JT, or better
SB: 22,87s,QTs,Kxs,Axs,Axo,Kxo,QT,JT or better

Ran it until I had 100,000 deals folded to the big blind. Results:

10.8% of deals folded to the big blind

Probability that the big blind:

Is suited: 23.3% (vs. random 23.5%)
Is paired: 5.9% (vs. random 5.9%)

Holds an ace: 18.5% (vs. random 14.9%)
king: 16.3%
queen: 15.7%
jack,ten: 15.5%
nine,eight: 14.4%
2-7: 13.6-14.1%

Specific hands:
AA: 139:1 (instead of 220:1)
KK: 177:1
QQ: 201:1
small pair: 250:1
AKo: 81:1 (instead of 110:1)
A2o: 95:1
72o: 123:1

Uncertainties in the above are about 0.1% for probabilities of suited/holding specific cards, about about 0.03% for specific hands (i.e., A2o between 92:1 and 98:1)

Would be happy to tweak the parameters a bit or extract particular bits of information for you if there's something else along these lines you're interested in.