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Poker Cards

July 3rd, 2010 admin No comments

Poker Cards

What are the chances, of recieving a hand of poker and all five cards are the same suit?

What are the chances, of recieving a hand of poker and all five cards are the same suit?
I saw the question on TV today and the answer was 25% but I believe its less as the first card drawn would be a twenty five then you have four more draws after so the percentage would diminish. Someone put it right for me. as I cant sleep.?

Here is the correct answer showing the math.

First we must determine the total number of possible 5 cards hands we can be dealt from a standard 52 Card Deck. To do this we will use the combination math function. The combination function makes use of factorials. The factorial function is defined as:

n! = n X (n - 1) X (n - 2) X ... X 1

For example:

6! = 6 X 5 X 4 X 3 X 2 X 1 = 720

The combination function which determines how many sets of r objects each (combinations) you can make from a defined number of n objects is defined as follows:

C(n,r) = n! / r! (n - r)!

So the number of 5 card hands which can be made from a single deck of 52 cards is as follows:

C(52,5) = 52! / (5! X (52 - 5)!)
= 52! / (120 X 47!)
= (52 X 51 X 50 X 49 X 48 X 47!) / (120 X 47!)
= (52 X 51 X 50 X 49 X 48) / 120
= 311875200 / 120
= 2598960 possible 5 card hands from a deck of 52 cards

Now that we know the number of all possible 5 card poker hands we can calculate the odds of being dealt a particular hand.

A flush is 5 cards all of the same suit. There are four suits of 13 cards each in a standard 52 card deck. To calculate the number of possible flush hands we can use the combination function from above as follows:

C(13,5) = 13! / (5! X (13 - 5)!
= 13! / (120 X 8!)
= (13 X 12 X 11 X 10 X 9 X 8!) / (120 X 8!)
= (13 X 12 X 11 X 10 X 9) / 120
= 154440 / 120
= 1287 possible hands of 5 cards all of the same suit for a single suit

We take the above number and multiply by 4 to get 5148 possible flush hands. It should be noted that this figure includes all straight flushes and royal flush hands also. We calculate the odds of being dealt a flush by dividing 5148 by 2598960 which gives us 0.0019808 or 0.19808%.

If you wish to eliminate the royal flush/straight flush hands, you should subtract 40 (the total number of straight flush hands) from 5148 which gives us 5108. So we divide 5108 by 2598960 which gives us 0.0019654 or 0.19654%.

To convert the above percentages to odds you would subtract the percentage value from 100 and divide the result by the percentage value:

Odds of being dealt a flush (including straight flush hands):
(100 - 0.19808) / 0.19808 = 99.80192 / 0.19808 = 503.8465-to-1 or 504-to-1

Odds of being dealt a flush (excluding straight flush hands):
(100 - 0.19654) / 0.0.19654 = 99.80346 / 0.19654 = 507.8022-to-1 or 508-to-1

I am not sure where they got the 25% figure but it is totally wrong.