what are the chances of being dealt 13 cards[reg deck]and receiving the queen of spades as the only spade.?

Question

The single spade[queen] occurred in hearts game where each of four players receive 13 cards. It's happened twice to me in the last 20 or so games.I think it's quite rare..Getting stuck with the queen of spades in the game of hearts is bad which is why I was aware of the above described two occurrences. If possible,can you please show how you arrived at the answer.Thank you very much.

Answer 1

C(39, 12) / C(52, 13) is the formula.

{where C(a, b) is the combination of a things taken b at a time and is computed: a!/[(a-b)!b!] }



There are C(52, 13) possible 13 card hands.

If you have the Q of spades, then there are 39 non-spades to choose 12 from. This is C(39, 12).



= 3910797436 / 635013559600



= 0.006158604610684915...



or 1 about in 162



(not as hard as getting pocket aces in texas hold-em!)



Note: after 113 hands the odds are about 50:50 that you get such a hand at least once. I don't play hearts, so I don't know how many hands an average game lasts, but what happened to you doesn't seem that unlikely.

Answer 2

You have a 1:4 chance of getting the ace of spades, so you want to know what is the chance that the other 12 cards are all not spades and multiply those together.



To figure the chance of getting 12 cards that are all not spades, you would multiply out 39/52 * 38/51 * 37/50...for 12 cards (last one is 28/41)...it keeps going down by one as you get one non-spade card each time.



In the game of hearts, people usually pass cards around, which can increase the chance you have no other spades (if you don't keep your low spades, or give away high spades).

Answer 3

1 out of 52, sure the person dealing is not crooked.