Probability of set OR overpair on flop

[Dex]

Say you're playing hold 'em and you have pocket JJ. What is the probability that you will either flop a set (J on flop) OR an overpair (no A, K, or Q on flop)?

And, how do you calculate the exact answer?

Thanks.

[BruceZ]

[ QUOTE ]
Say you're playing hold 'em and you have pocket JJ. What is the probability that you will either flop a set (J on flop) OR an overpair (no A, K, or Q on flop)?

And, how do you calculate the exact answer?

Thanks.

[/ QUOTE ]

P(at least 1 J on flop) = 1 - P(no J on flop)
= 1 - (48/50 * 47/49 * 46/48)
=~ 11.76%,

or we can compute this as 1 - C(48,3)/C(50,3) =~ 11.76%.

Next we need

P(no J,Q,K,A) = 36/50 * 35/49 * 34/48 =~ 36.43%,

or we can compute this as C(36,3)/C(50,3) =~ 36.43%.

Adding these two probabilities gives about 48.2%.

Note that this includes full houses and quads. If you wish to exclude jacks full over Q,K, or A, then subtract off

P(JQQ,JKK,or JAA) = 2/50 * 12/49 * 3/48 * 3 =~ 0.18%,

or compute this as 2*3*C(4,2)/C(50,3) =~ 0.18%.

To exclude ALL full houses with 3 jacks, subtract 2/50 * 48/49 * 3/48 * 3 =~ 0.73%,

or compute this as 2*12*C(4,2)/C(50,3) =~ 0.73%.

The probability of quad jacks is just 48/C(50,3) =~ 0.25%.

The above probability for P(no J,Q,K,A) includes cases where the board pairs or trips to give you 2-pair or a full house, which could give someone else trips or quads. To exclude these flops, recompute this as 36/50 * 32/49 * 28/48 =~27.43%.