# Probability of a winning hand in the card game Sevens

I'm trying to calculate the probability of being dealt a winning hand from the start in the card game "Sevens" while playing as 4 players.

By being 4 players you get 13 cards each, so using the formula for combinations: C(n,r) = n! / r!(n-r)! where n = 52 and r = 13 I get a total of 635.013.559.600,00 different combinations of hands.

To win from the very start I could be dealt an entire suit, which gives me 4 different possibilites of a winning hand. So that alone should give me 1 in a roughly 159 billion chance to win right away.

I could also be given all the 7s, 8s, 9s and one of the 10s to win. So my question is: how do I figure out how many different hands there are that can win the game right away?

• It is not perfectly clear what the exact conditions for immediately winning is, but as for counting the number of hands containing all of the 7s, 8s, 9s and one of the 10s.. this too can be accomplished in exactly four ways noting that the only piece of unknown information about the hand is which suit the 10 is from. – JMoravitz Mar 15 '18 at 19:44
• If I am interpreting the rules correctly, a turn can be broken into phases as follows. p1) Play the 7 of hearts if it is in your hand and skip to second phase. Otherwise play a card of rank 1 higher than the top card of the discard pile. If unable, then immediately end your turn. p2) Having played a card in phase 1, you may continue to play additional cards of the same rank as the card played in phase 1. Once satisfied, continue to next phase. p3) You may play an additional card of the same suit as the most recently played card so long as they are of adjacent rank. p3 may be repeated. – JMoravitz Mar 15 '18 at 19:53
• Assuming that is all correct, note that it is possible to lose even if you are dealt all of the spades so long as your opponent was dealt an immediate win hand and holds the seven of hearts since his turn will come before yours. There would also be the possibility of having an immediate win hand by conditions other than the two you mention, e.g. having the four sevens and then having a run of cards of a suit going either up or down from seven, having three sevens (including the 7 of hearts) and a run going up or down from a suit of a seven you hold, etc... – JMoravitz Mar 15 '18 at 19:57