5-of-a-kind:
13C1 * 4C4 * 1 = 13
probability = 13/2869685 = .00000453
Three-of-a-kind when ranked ABOVE Two Pair:
(This can either be dealt naturally, as above, or with one pair plus the wild joker)
54912 + 13 C1 * 4 C2 * 1 * 12 C2 * 4 C1 * 4 C1 = 54912 + 13 * 6 * 66 * 4 * 4 = 137280
Probability = 137,280/2,869,685 = .04784
Two Pair when ranked BELOW Three of a Kind:
(Two pair can only be dealt naturally, because if one were dealt One Pair plus a joker, it would be declared as Three of a Kind not Two Pair.)
As calculated above, there are 123,552 get be dealt Two Pair.
Probability = 123,552/2,869,685 = .043054
Note that this probability is LESS than the probability of being dealt Three of a Kind.
When Two Pair is ranked ABOVE Three of a Kind:
(Using the same calculations as above)
123,552 + 13 C1 * 4 C2 * 1 * 12 C2 * 4 C1 * 4 C1 = 205,920
Probability = 205,920/2,869,685 = 0.071757
Three-of-a-kind when ranked BELOW Two Pair:
(This hand can only be dealt naturally as one pair plus a wild card would be declared as Two Pair.)
Using the calculation above, number of ways = 5,4912
Probability = 5,4912/2,869,685 = 0.0191352
Note that this probability is now LOWER than Two Pair, despite (and because of) being ranked lower in the hierarchy of hands.
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