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A) What is the probability that when you are holding Ax that someone is holding Ay (where y>x) with n opponents.
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I generated some probability tables for the case where you hold Ax and the flop contains a single A. There are three tables, one for the case where the two other cards on the flop are of smaller rank than your kicker, a second for the case where one of the flop cards is higher than your kicker, and the last where two cards on the flop are higher than your kicker.
Unfortunately the formatting was lost, so the tables are kind of messy...
Examples:
1) you hold AQ and the flop comes AT7 against 9 opponents. In this case there are two overcards to your kicker (A, K), the flop contains 0 higher cards than your kicker (we exclude the A). So there is about a 7.2% chance that at least one of your opponents is holding AA or AK.
2) You hold AJ and the flop comes AQ2 against 6 opponents. There are 3 higher kickers (A,K,Q) than your J, and the flop contains 1 card higher than your kicker. Using table 2 the probability of at least one opponent holding AA, AK, or AQ is 8.0%.
3) You hold A4 and the flop comes A85 against 7 opponents. There are 10 higher cards than your kicker, and the flop contains two cards higher than your kicker. Using table 3, the probability of at least one opponent holding A5-AA is 37%.
<font class="small">Code:</font><hr /><pre>
Table for 0 higher cards than your kicker on the flop
# cards higher than your kicker
1 2 3 4 5 6 7 8 9 10 11 12
Opponents
1 0.1 0.8 1.6 2.3 3.1 3.8 4.5 5.3 6.0 6.8 7.5 8.2
2 0.2 1.7 3.1 4.6 6.0 7.4 8.9 10.3 11.7 13.0 14.4 15.8
3 0.3 2.5 4.6 6.8 8.9 11.0 13.0 15.0 17.0 18.9 20.8 22.7
4 0.4 3.3 6.1 8.9 11.7 14.3 16.9 19.5 22.0 24.4 26.8 29.1
5 0.5 4.1 7.6 11.0 14.4 17.6 20.7 23.7 26.7 29.5 32.3 34.9
6 0.6 4.9 9.1 13.1 17.0 20.7 24.3 27.7 31.1 34.3 37.3 40.3
7 0.6 5.7 10.5 15.1 19.5 23.7 27.7 31.6 35.2 38.7 42.0 45.2
8 0.7 6.5 11.9 17.1 22.0 26.6 31.0 35.2 39.1 42.8 46.4 49.7
9 0.8 7.2 13.3 19.0 24.3 29.4 34.1 38.6 42.8 46.7 50.4 53.8
Table for 1 higher cards than your kicker on the flop
# cards higher than your kicker
2 3 4 5 6 7 8 9 10 11 12
Opponents
1 0.6 1.4 2.1 2.9 3.6 4.3 5.1 5.8 6.6 7.3 8.0
2 1.3 2.8 4.2 5.7 7.1 8.5 9.9 11.3 12.7 14.1 15.4
3 1.9 4.1 6.2 8.4 10.4 12.5 14.5 16.5 18.4 20.4 22.3
4 2.6 5.4 8.2 11.0 13.7 16.3 18.9 21.4 23.8 26.2 28.5
5 3.2 6.7 10.2 13.5 16.8 19.9 23.0 25.9 28.8 31.6 34.3
6 3.8 8.0 12.1 16.0 19.8 23.4 26.9 30.3 33.5 36.6 39.6
7 4.4 9.3 14.0 18.4 22.7 26.7 30.6 34.3 37.8 41.2 44.4
8 5.1 10.6 15.8 20.8 25.5 29.9 34.1 38.1 41.9 45.5 48.9
9 5.7 11.8 17.6 23.0 28.2 33.0 37.5 41.7 45.7 49.5 53.0
Table for 2 higher cards than your kicker on the flop
# cards higher than your kicker
3 4 5 6 7 8 9 10 11 12
Opponents
1 1.2 1.9 2.7 3.4 4.2 4.9 5.6 6.4 7.1 7.9
2 2.4 3.8 5.3 6.7 8.2 9.6 11.0 12.4 13.7 15.1
3 3.6 5.7 7.8 9.9 12.0 14.0 16.0 18.0 19.9 21.8
4 4.7 7.5 10.3 13.0 15.6 18.2 20.7 23.2 25.6 27.9
5 5.9 9.3 12.7 16.0 19.2 22.2 25.2 28.1 30.9 33.6
6 7.0 11.1 15.1 18.9 22.5 26.0 29.4 32.7 35.8 38.8
7 8.1 12.8 17.3 21.6 25.7 29.7 33.4 37.0 40.4 43.6
8 9.2 14.5 19.6 24.3 28.8 33.1 37.2 41.0 44.6 48.1
9 10.3 16.2 21.7 26.9 31.8 36.4 40.7 44.8 48.6 52.1
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Explainer:
First you calculate the probablity of a single opponent holding Ay (where y is a overcard to your kicker). Let's consider the case where the flop contains no overcards to your kicker. On the flop, we've seen 5 of 52 cards, leaving 47 unseen. The possible starting hands are C(47,2)=1081. If there are N overcards to your kicker (eg. if you hold A8, there are 6 cards higher than 8, so N=8), then there are 4*N cards in the deck higher than your kicker. That makes a total of 2*4*(N-1)+1 possible starting hands which include one of the two remaining Aces (remember, there is one on the flop). The N-1 factor counts all the cards up to K. The +1 factor is for the single possibility of an opponent holding pocket Aces. That leaves us with the probablity p(n)=(2*4*(N-1)-1)/1081
To get the probability of at least 1 of K opponents holding and A with a better kicker, we subtract the probably of no opponent holding a better kicker from 1: P(N,K) = 1 - (1 - p(N))^K.
Where there are overcards to your kicker on the flop, we must subtract 2 starting hands for each (matched with each of the two remaining Aces), so the probability function becomes: p'(n,o) = (2*(4*(N-1) - o))/1081, where o is the number of overcards and n is the number of cards of higher rank in a single suit.