Re: 4 of a kind
C(48,3)/C(50,5) = 17296/2118760 = 0.0081633
Squaring the above yields 0.0000066639. Of course, there
are less likely scenarios such as having quads on board or
making quads with just one card.
It is actually MORE likely that you will make quads by using just 1 card from your hand than with a pair in the hole when you factor in the probability of getting dealt a non-pair which is 16 times the probability of getting dealt a pair. You can also get a pair on one round and a non-pair on the other.
[ (1/17)*C(48,3) + (16/17)*2*C(47,2) ]^2 / C(50,5)^2
= 481,866 to 1
This includes everything except the split pot with quads on board.
If we only consider pairs in the hole, then it is (1/17)^2 to get dealt a pair twice in a row, times 1/1,5006.25 to make quads after we get the pair, or 4,336,805 to 1 all together to get dealt a pair and make quads twice in a row. So it is much less likely to do it with a pairs alone than with non-pairs by about a factor of 10.
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