A:
Your chances of being dealt four to a royal and drawing the fifth card are one in 47 (1 in 48 if it's a Joker game).
Your chances of being dealt a royal are once every 649,740 hands.
And your chances of being dealt a sequential royal (meaning that the five cards line up in order, forward or backward: 10, jack, queen, king, ace or ace, king, queen, jack, ten) are one in 38,984,400. (You'd have to play 500 hands an hour, 24 hours a day, for 3,248.7 days to hit one, theoretically.)
Update 03 July 2006
Today’s question remains as it was originally posted, but we received the following early response from Stewart Ethier, one of the world’s top gambling mathematicians.
“Today's QoD answer correctly answers a question that wasn't asked. The question ‘What are my chances of being dealt four to a royal and drawing the fifth card?’ asks for P(A and B), while the answer gives us P(B given A). Sorry to nitpick, but I believe that my interpretation is the one the questioner intended."
(Editor’s note: Stew is correct. The literal wording of the question does not ask for the probability of making a royal given that four to the royal have already been dealt, but the chances of beginning a new hand, being dealt four to a royal, then completing it. The rest of Stew’s response, which gives the correct answer to the question of 1 in 129,948, follows.)
“To answer correctly, there are four royals, five ways in which one of the five cards can be missing, and 47 ways in which the fifth non-royal card can be chosen. That gives P(A) = (4)(5)(47)/2,598,960.
Now multiply this by [today’s] answer P(B given A) = 1/47 to get P(A and B) = P(A) P(B given A) = (4)(5)/2,598,960 = 1/129,948.”
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