扑克牌

zolo

一副52张的扑克牌，从里面抽出五张牌，那么得到
a.顺子
b.同花
c.full house
d.四张一样的牌
e.同花顺
的几率是多少？

JamesTea

想尝试解答，虽然不是很确定答案是否正确。。。

since we are now considering the arrangement and order of the cards, concept of Permutation is used:

To do permutation of 5 cards out of 52... hence there are 52P2 = 311875200 ways to arrange.

b) to get same shapes:
    _   _   _   _   _
  for the 1st card there are 13 ways
  for the 2nd card there are 12 ways
  for the 3rd card there are 11 ways
  for the 4th card there are 10 ways
  for the 5th card there are 9 ways
  so there are total 13 x 12 x 11 x 10 x 9 x 4 (there are 4 shapes) = 617760 ways to arrange cards of same shapes,

  Hence, P(getting 5 same shapes cards) = 617760 / 311875200 = 33 / 16660


c) to get a full house, we divide into 2 groups
     _   _   _                                                                _   _
   there are 4 ways to get the 1st card                 there are 4 ways to get the 1st card
   there are 3 ways to get the 2nd card                there are 3 ways to get the 2nd card     
   there are 2 ways to get the 3rd card

  1st groups there are total 4x3x2x13(13 possible numbers) = 312 ways
  2nd group there are total 4x3x12(since one of the shape is taken by 1st group) = 144 ways
  so total ways for the full house to be arranged is 312 x 144 = 44928 ways

  Hence, P(to get a full house) = 44928 / 311875200 = 3 / 20825

d) to get 4 same number cards, also we divided into 2 groups
     _   _   _   _                                                               _
   there are 4 ways to get the 1st card                 there are 48 ways to get the 1st card
   there are 3 ways to get the 2nd card                  
   there are 2 ways to get the 3rd card
   there are 1 way to get the 4th card
   so for the 1st group there are 4x3x2x1x13 (13 possible numbers) = 312 ways
   for the 2nd group there are 48 ways to arrange anyone card
   therefore total possible arrangement of 4 same number cards = 312 x 48 = 14976 ways

   Hence, P(getting 4 same numbered cards) = 14976 / 311875200 = 1 / 20825

e) to get a 同花顺,
     _   _   _   _   _
     there is only 1 way to arrange each time, i.e.  A 2 3 4 5 , 2 3 4 5 6, ...
     so there are 9 ways to arrange 同花顺 for 1 shape, since there are 4 possible shapes.
     total ways of arrange 同花顺 = 36 ways.

     Hence, P(getting 同花顺) = 36 / 311875200 = 0.0000001154308

第一的不是很会，再尝试下

50912cmea

回复 2# JamesTea

大大，

A) 顺子 (straight)
9 scenarios:
1) 2 3 4 5 6
2) 3 4 5 6 7
3) 4 5 6 7 8
4) 5 6 7 8 9
5) 6 7 8 9 10
6) 7 8 9 10 J
7) 8 9 10 J Q
8) 9 10 J Q K
9) 10 J Q K 1

For 1st row, 1st number has 20 可能 (=4 different shapes * 5 numbers can be chosen, i.e. 2,3,4,5,6)
For 1st row, 2nd number has 16 可能
(=4 different shapes * 4 numbers can be chosen, i.e. 3,4,5,6)
"
"
For 1st row, 5th number has 4 可能
(=4 different shapes * 1 numbers can be chosen, i.e. 6)

=5!*4^5/52P5 = 0.000394003 (1st scenario)
all 9 scenarios = 0.000394003*9 = 0.003546033 = 192/54145 (the possibility includes 同花顺的)


C) 一对与三条 (full house)
Hence, P(to get a full house) = 44928/311875200*5C2 (order fully arranged) = 0.001440576 = 6/4165


D) 四条 (four of a kind)
Hence, P(getting 4 same numbered cards) = 14976/311875200*5C1 (order fully arranged) = 0.000240096 = 1/4165


E) 同花顺 (royal flush)
Hence, P(getting 同花顺) = 36/311875200*5! (1st number can be chosen from 5 numbers, i.e. 2,3,4,5,6; 2nd number has 4 numbers...) = 0.000013851 = 3/216580

请看 http://www.stat.nuk.edu.tw/prost/simulation/Poker/Poker.htm