Poker Project I need help fast

Question

How many different hands can you have that are all spades? hearts? diamond? clubs?

What is the probability of getting a flush in spades?

What is the probability of getting a flush in any suit?

Answer 1

The number of possible poker hands (5 card) that can be a flush (cards all of the same suit) is

₁₃C₅ = 13! / (13 - 5)!(5!)

₁₃C₅ = 1287

The number of 5 card hands is

₅₂C₅ = 52! / (52 - 5)!(5!)

₅₂C₅ = 2568960

Probability of getting a flush in spades - and any suit - is

1287 / 2568960

0.000495

0.0495%

Answer 2

I'm assuming from the question that a hand consists of five cards.

Note that a deck of cards has the same number of spades, hearts, diamonds, and clubs, so once we find the number of different hands that are all spades, we know the answer for the rest of the suits.

There are 13 spades in the deck. We need to know the number of ways to create hands of 5 cards that are all spades. Using the combination formula, we will choose 5 cards from the 13 spades:

_nC_r = ( n! ) / ( r! × (n - r)! )

₁₃C₅ = ( 13! ) / ( 5! × 8! )

₁₃C₅ = 1,287

So, there are 1,287 possible hands that are all spades. This number is the same for hands of all hearts, diamonds, and clubs.

Now we need to find the probability of getting a flush in spades. There are 1,287 possible hands that are all spades. You could consider this the number of possible flushes of spades. However, this includes the possible straight flushes in spades (this is a hand of all spades that are also in consecutive order, e.g. 5, 6, 7, 8, 9, in spades). There are 10 possible straight flushes in spades. The question probably wants you to exclude the straight flushes from the probability of ordinary flushes. So, there are really 1,287 - 10 = 1,277 possible flushes in spades. [NOTE: if the question wants you to include the straight flushes, use 1,287 instead for the remainder of this problem]

We calculate the probability of a flush of spades by dividing the number of possible spade flush hands by the number of all possible 5-card hands.

The total number of 5-card hands is:

_nC_r = ( n! ) / ( r! × (n - r)! )

₅₂C₅ = ( 52! ) / ( 5! × 47! )

₅₂C₅ = 2,598,960

The probability of a flush of spades is:

1,277 / 2,598,960 = 0.00049135 OR 0.049135%

The probability of a flush of any suit is simply the probability of a flush of spades, times 4 (since there are 4 suits and every suit has 13 cards, and therefore an equal probability of creating a flush):

4 × 0.00049135 = 0.0019654 OR 0.19654%