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I just realized you meant that you had AT, not you had "a 10" -- caps lock is confusing.
To adjust for that:
Oppo now had nCr(50,2) = 1225 possible holdings
Oppo cards you're worried about (no longer including AA):
AK - 12 cards
AQ - 12 cards
AJ - 12 cards
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total: 36 cards
1 - (1189/1225)*(1188/1224)*(1187/1223)=.0859
So final answer is closer to 8%
ALSO NOTE: This is a VERY CLOSE approximation (within a percent for sure), but it is NOT EXACT. The reason it's not exact is that I'm assuming that the three players not having the cards in question are independent events, when in fact they are LOOSELY dependent.
gm
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Even assuming independence, the denominator would not just change by 1 for each player. The first one would be C(50,2) = 1225, but the second should be C(48,2) = 1128, and the third should be C(46,2) = 1035. The numerator would change by the same amount, but only if we assume that an A,K,Q, or J was not dealt. It is not possible to get an exact answer by this method easily, but we can get an exact answer by the
inclusion-exclusion principle as:
3*36/1225 -
C(3,2)*36*30 / C(50,2) / C(48,2) +
C(3,3)*36*(6*25+24*24) / C(50,2) / C(48,2) / C(46,2)
=~ 8.58%.