I have derived the exact probabilities of two or more players being dealt a pocket pair among N players for N=2 to 9. Of course, for small values of N these probabilities can be derived fairly easily.
For large values of N these probabilities are difficult to derive and heretofore have been estimated using an assumption of independence across the players so that the binomial formula can be utilized.
Below I will give the exact number of deals for each N that have P players being dealt a pocket pair for P=0 to N. I will give the exact probabilities of each pair occurring and compare it to the probability approximated from using the independent binomial assumption (and give their ratio too).
I will report probabilities as percentages, giving four decimal places in most cases. Since some of these probabilities are quite small, I will always give at least two significant digits. As usual, displayed probabilities may on occasion not appear to add to one due to rounding.
For N players, the total number of deals consisting of each player receiving two cards is easy to calculate. I will use the C(X,Y) notation for the number of combinations of choosing Y items from X items, where the order is irrelevant. There are C(52, 2N) ways of choosing the 2N cards to be dealt to the N players. Then the number of ways these 2N cards can be dealt to the N players to form N 2-card hands is the indexed product of C(2J, 2) with index J going from N to 1. [For example, for N=3 players and 2N=6 cards, the first player can get C(6,2) hands, the second player C(4,2) hands, and the third player C(2,2) hands.] We then need to divide by N! (N factorial) since we are not interested in which player gets which hand, only the overall combination of N 2-cards hands.
As an aside, this last term simplifies to the product of the odd numbers up to 2N-1, or equivalently the product of the first N odd numbers, which using the double factorial notation is (2N-1)!!. So for N=3, this last term is 1*3*5 = 15, meaning that there are 15 ways to deal a given 6 cards to form 3 2-card hands where the order of the cards in each player's hand is irrelevant [(As, Kh) is an equivalent hand to (Kh, As)] and the order of the players' hands is irrelevant [the deal of {(As, Kh), (Qd, Jc), (6c, 3h)} is equivalent to {(Qd, Jc), (6c, 3h), (As, Kh)}].
So the total number of deals of N players each receiving two cards is C(52, 2N) * (2N-1)!!.
Let's start easy with N=1. Of course there are C(52,2) = 1,326 ways of one player being dealt two cards. 1,248 of these will result in no pair and the other 78 ways will result in a pair. Clearly, after any one of the 52 cards is first dealt to the player, 48 of the remaining 51 cards will give no pair and 3 of the remaining 51 cards will give one pair (we then need to divide by two in both cases since the order of the cards in the hand is irrelevant). I will record this as follows:
N=1 Player
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No Pair : 1,248 ( 94.1176%)
One Pair: 78 ( 5.8824%)
TOTAL DEALS: 1,326 (100%)
where the exact percentages are given in parentheses. In this case, I do not display the approximate probabilities given by the independent binomial assumption since the two cases are identical (there is only one player).
For two players, there are C(52,4) * (1*3) = 812,175 ways of dealing two players two-card hands.
N=2 Players
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No Pair : 719,472 ( 88.5858%; 88.5813%; 1.000051)
One Pair: 89,856 ( 11.0636%; 11.0727%; 0.999184)
Two Pair: 2,847 ( 0.3505%; 0.3460%; 1.013061)
TOTAL DEALS: 812,175 (100%)
where now in parentheses are shown the exact probability, the approximate probability using the independent binomial assumption, and the ratio of the exact probability to the approximate probability.
You will see that the independent binomial approximation underestimates the probability of 0 pair and 2 pair, and overestimates the probability of 1 pair. This tendency holds for all N as in reality one player holding a pocket pair makes it slightly more likely that any other player also holds a pocket pair, and one player not holding a pocket pair makes it slightly more likely that any other player also does not hold a pocket pair.
For three players, there are C(52,6) * (1*3*5) = 305,377,800 ways of dealing three players two-card hands.
N=3 Players
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No Pair : 254,634,432 ( 83.3834%; 83.3706%; 1.000153)
One Pair: 47,661,120 ( 15.6073%; 15.6320%; 0.998418)
Two Pair: 3,017,664 ( 0.9882%; 0.9770%; 1.011437)
Three Pair: 64,584 ( 0.0211%; 0.0204%; 1.039045)
TOTAL DEALS: 305,377,800 (100%)
For four players, there are C(52,8) * (1*3*5*7) = 79,016,505,750 ways of dealing four players two-card hands.
N=4 Players
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No Pair : 62,020,470,096 ( 78.4905 %; 78.4665 %; 1.000306)
One Pair: 15,464,756,736 ( 19.5716 %; 19.6166 %; 0.997702)
Two Pair: 1,467,494,496 ( 1.8572 %; 1.8391 %; 1.009865)
Three Pair: 62,764,416 ( 0.0794 %; 0.0766 %; 1.036600)
Four Pair: 1,020,006 ( 0.00129%; 0.00120%; 1.078153)
TOTAL DEALS: 79,016,505,750 (100%)
For five players, there are C(52,10) * (1*3*5*7*9) = 14,949,922,887,900 ways of dealing five players two-card hands.
N=5 Players
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No Pair : 11,046,277,262,592 ( 73.8885 %; 73.8508 %; 1.000511)
One Pair: 3,439,978,397,856 ( 23.0100 %; 23.0784 %; 0.997037)
Two Pair: 434,873,140,416 ( 2.9089 %; 2.8848 %; 1.008343)
Three Pair: 27,876,790,656 ( 0.1865 %; 0.1803 %; 1.034209)
Four Pair: 905,389,056 ( 0.0061 %; 0.0056 %; 1.074858)
Five Pair: 11,907,324 ( 0.000080%; 0.000070%; 1.130889)
TOTAL DEALS: 14,949,922,887,900 (100%)
For six players, there are C(52,12) * (1*3*5*7*9*11) = 2,145,313,934,413,650 ways of dealing six players two-card hands.
N=6 Players
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No Pair : 1,492,278,187,226,112 ( 69.5599 %; 69.5067 %; 1.000766)
One Pair: 557,175,599,735,040 ( 25.9718 %; 26.0650 %; 0.996423)
Two Pair: 87,971,700,939,408 ( 4.1006 %; 4.0727 %; 1.006872)
Three Pair: 7,512,990,046,848 ( 0.3502 %; 0.3394 %; 1.031871)
Four Pair: 365,736,653,568 ( 0.0170 %; 0.0159 %; 1.071618)
Five Pair: 9,613,334,016 ( 0.00045 %; 0.00040 %; 1.126693)
Six Pair: 106,478,658 ( 0.0000050%; 0.0000041%; 1.198023)
TOTAL DEALS: 2,145,313,934,413,650 (100%)
For seven players, there are C(52,14) * (1*3*5*7*9*11*13) = 239,049,266,977,521,000 ways of dealing seven players two-card hands.
N=7 Players
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No Pair : 156,549,075,187,355,136 ( 65.4882 %; 65.4180 %; 1.001073)
One Pair: 68,133,459,724,881,408 ( 28.5018 %; 28.6204 %; 0.995858)
Two Pair: 12,898,104,722,021,376 ( 5.3956 %; 5.3663 %; 1.005453)
Three Pair: 1,375,801,040,877,120 ( 0.5755 %; 0.5590 %; 1.029586)
Four Pair: 89,232,018,258,240 ( 0.0373 %; 0.0349 %; 1.068434)
Five Pair: 3,515,707,001,664 ( 0.0015 %; 0.0013 %; 1.122557)
Six Pair: 77,831,088,192 ( 0.000033 %; 0.000027 %; 1.192862)
Seven Pair: 746,037,864 ( 0.00000031%; 0.00000024%; 1.280607)
TOTAL DEALS: 239,049,266,977,521,000 (100%)
For eight players, there are C(52,16) * (1*3*5*7*9*11*13*15) = 21,006,454,335,649,657,875 ways of dealing eight players two-card hands.
N=8 Players
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No Pair : 12,952,160,151,935,374,656 ( 61.6580 %; 61.5699 %; 1.001431)
One Pair: 6,436,718,641,227,663,360 ( 30.6416 %; 30.7850 %; 0.995344)
Two Pair: 1,420,395,848,998,993,152 ( 6.7617 %; 6.7342 %; 1.004084)
Three Pair: 181,664,175,195,689,472 ( 0.8648 %; 0.8418 %; 1.027354)
Four Pair: 14,716,813,939,542,000 ( 0.0701 %; 0.0658 %; 1.065306)
Five Pair: 772,570,615,474,944 ( 0.0037 %; 0.0033 %; 1.118482)
Six Pair: 25,638,362,624,160 ( 0.00012 %; 0.00010 %; 1.187766)
Seven Pair: 491,218,535,808 ( 0.0000023 %; 0.0000018 %; 1.274393)
Eight Pair: 4,155,760,323 ( 0.000000020%; 0.000000014%; 1.380032)
TOTAL DEALS: 21,006,454,335,649,657,875 (100%)
For nine players, there are C(52,18) * (1*3*5*7*9*11*13*15*17) = 1,470,451,803,495,476,051,250 ways of dealing nine players two-card hands.
N=9 Players
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No Pair : 853,667,643,003,891,797,796 ( 58.0548 %; 57.9481 %; 1.001840)
One Pair: 476,852,107,923,575,077,980 ( 32.4290 %; 32.5958 %; 0.994880)
Two Pair: 120,157,940,672,756,034,048 ( 8.1715 %; 8.1490 %; 1.002766)
Three Pair: 17,914,600,053,357,815,808 ( 1.2183 %; 1.1884 %; 1.025174)
Four Pair: 1,740,207,513,284,368,128 ( 0.1183 %; 0.1114 %; 1.062234)
Five Pair: 114,111,043,097,923,872 ( 0.0078 %; 0.0070 %; 1.114466)
Six Pair: 5,045,885,892,919,872 ( 0.00034 %; 0.00029 %; 1.182736)
Seven Pair: 144,930,110,637,312 ( 0.0000099 %; 0.0000078 %; 1.268252)
Eight Pair: 2,450,932,035,552 ( 0.00000017 %; 0.00000012 %; 1.372645)
Nine Pair: 18,577,440,882 ( 0.00000000126%; 0.00000000084%; 1.498219)
TOTAL DEALS: 1,470,451,803,495,476,051,250 (100%)
From these tables, it is straightforward to calculate any other probability of interest, for example the probability of two-or-more players being dealt a pocket pair among N players. The table below gives the exact and approximate probabilities and their ratio for N=2 to 9.
Exact Prob Approx Prob
N 2+ Pairs (%) 2+ Pairs (%) Ratio
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2 0.3505 0.3460 1.013061
3 1.0093 0.9974 1.012001
4 1.9379 1.9169 1.010976
5 3.1015 3.0708 1.009987
6 4.4684 4.4284 1.009032
7 6.0099 5.9616 1.008111
8 7.7004 7.6451 1.007225
9 9.5163 9.4560 1.006371
As mentioned above, the independence binominal approximation underestimates the likelihood of multiple players being dealt pocket pairs.