Perms and Combs! and some induction..

careless25 · 2011-05-30 05:37:49

The answer i posted is correct...i verified with Teachers Assistant.
You consider the third case i posted because you can have one card which of same suit at lets say card 2 and can also be the same value as card 1. So yea three cases, add them together and divide by total possibilities.

bobbym · 2011-05-30 06:51:41

Hi careless25;

You might be right, I do not know yet. I am pretty sure mine is wrong.

Hi gAr;

I am getting his 10296 and 102960 but I am getting 51480 instead of his 41184 for one suit and one rank the same.

gAr · 2011-05-30 15:25:11

Hi bobbym,

I do not understand why combinations are used here.
I think permutation's appropriate.

bobbym · 2011-05-30 15:48:48

Hi gAr;

I agree, but as long as you use the same for both numerator and denominator the probability remains the same.

For instance for a badugi:

Is everything okay?

gAr · 2011-05-30 16:26:11

Hi,

Yes, I understand.
But when we use the way which is natural, there will be less confusion.

For careless25's 3 cases, I get:
52*36*22*(30)
52*36*22*(3)
52*36*22*(6)

bobbym · 2011-05-30 16:40:57

Hi;

So we agree on the 51480. It is a tricky problem.

gAr · 2011-05-30 17:05:23

Hi bobbym,

That's strange, your probability would go above 60%!
Is the denominator 52C4 ?

Hi careless25,

For M.I.,

bobbym · 2011-05-30 17:13:43

Hi gAr;

Yes, that is what the program is getting.

gAr · 2011-05-30 17:19:58

Hi,

So we agree on the 51480

If we agree on the numerator, what's going wrong with the final answer?!

bobbym · 2011-05-30 17:21:38

Hi;

What do you get for a probability on yours?

gAr · 2011-05-30 17:25:02

Much less!

bobbym · 2011-05-30 17:38:21

I know I had that too, I think I tried every answer so far.

gAr · 2011-05-30 17:42:01

Okay.
You mentioned about the program, what does it do?

bobbym · 2011-05-30 19:10:11

Takes all possible 4 permutations of 52. There are only 6 497 400 ways to arange them. Then it looks for 3 patterns {3,3} which means 3 different suits and 3 different ranks.{4,3} which means 4 different ranks and 3 different suits and last {3,4} which means 3 different ranks and 4 different suits.

Takes the whole sample space and checks them all for the conditions badugi demands.

gAr · 2011-05-30 20:11:34

Hi bobbym,

Is the answer in post #52 the output of your program?

bobbym · 2011-05-30 20:17:39

Yes, I finally got it to run. This turned out to be a difficult program to write in the functional style. The last few have been like that.

That is the output it is getting. But programs are written by the person solving the problem, they reflect his biases so on simulations they are sometimes unreliable.

gAr · 2011-05-30 20:53:32

Okay.
I wrote a procedural program, since only counting is required.

Anyway, I believe it must be less than 35%.

P(picking up the 4th card, which results in badugi or not) = 52*36*24*49 / (52*51*50*49)

So our answer must be less than that.

bobbym · 2011-05-30 20:58:40

Yes, but what about picking up a second card that ruins a badugi?

Does that not assume that you choose three good cards and then a bad fourth one. How about First a good one then a bad one followed by two good ones.

gAr · 2011-05-30 21:06:44

Hmmm, I understand why we are getting different values.
I'll check.

gAr · 2011-05-31 01:45:09

Hi bobbym,

After correcting my program:

which is the answer careless25 too got.

bobbym · 2011-05-31 09:35:42

Hi gAr;

What did you get for each case. On another forum they agree with my answer of 51480 instead of his 41184.

gAr · 2011-05-31 15:16:47

Hi bobbym,

1. diff value, same suit as one card : 2471040

2. diff suit, same value as one card : 247104

3. same suit, same value with another card : 988416

which agrees...

bobbym · 2011-05-31 16:25:49

Hi gAr;

1) 2471040 Kayrect!

2) 247104 Kayrect!

3) 988416 This is the one that is in dispute. 1 235 520 is what I am getting.

gAr · 2011-05-31 16:37:04

Hmmm, perhaps you overcounted something?

Did you understand the OP's solution?

bobbym · 2011-05-31 17:09:44

Hi gAr;

Yes, but I never agreed with that 3rd answer, the other two yes. I posted that one particular part on another forum. This was the reply.

ryanstone wrote:

First we calculate in how many ways can we have exactly 2 same ranks & suits .
With 4 suits we can have exactly 2 same in
[math]4 \binom{3}{2} =12 ways and with the13 ranks we can have exactly 2 same in
13 \binom{12}{2}=858 ways.
Together we can have exactly 2 same ranks & suits in 12 * 858= 10296 ways.
Now for a specific set among these....for example if among the 4 cards we have ranks K,K,J & 2,and suits s,s,d &h...let's see in how many ways can we distribute these ranks among the suits...
If both s are K,then no. ways the rest can be distributed is 2!=2
If no s are K, then no. ways the rest can be distributed is 1
If exactly one s is K the no. ways the rest can be distributed is 2!=2.
These are the only cases and hence all-together there are 5 ways for a definite of choice suits & ranks.
So total no. of possible winning arrangements is 5*10296=51480

Math Is Fun Forum

#51 2011-05-30 05:37:49

Re: Perms and Combs! and some induction..

#52 2011-05-30 06:51:41

Re: Perms and Combs! and some induction..

#53 2011-05-30 15:25:11

Re: Perms and Combs! and some induction..

#54 2011-05-30 15:48:48

Re: Perms and Combs! and some induction..

#55 2011-05-30 16:26:11

Re: Perms and Combs! and some induction..

#56 2011-05-30 16:40:57

Re: Perms and Combs! and some induction..

#57 2011-05-30 17:05:23

Re: Perms and Combs! and some induction..

#58 2011-05-30 17:13:43

Re: Perms and Combs! and some induction..

#59 2011-05-30 17:19:58

Re: Perms and Combs! and some induction..

#60 2011-05-30 17:21:38

Re: Perms and Combs! and some induction..

#61 2011-05-30 17:25:02

Re: Perms and Combs! and some induction..

#62 2011-05-30 17:38:21

Re: Perms and Combs! and some induction..

#63 2011-05-30 17:42:01

Re: Perms and Combs! and some induction..

#64 2011-05-30 19:10:11

Re: Perms and Combs! and some induction..

#65 2011-05-30 20:11:34

Re: Perms and Combs! and some induction..

#66 2011-05-30 20:17:39

Re: Perms and Combs! and some induction..

#67 2011-05-30 20:53:32

Re: Perms and Combs! and some induction..

#68 2011-05-30 20:58:40

Re: Perms and Combs! and some induction..

#69 2011-05-30 21:06:44

Re: Perms and Combs! and some induction..

#70 2011-05-31 01:45:09

Re: Perms and Combs! and some induction..

#71 2011-05-31 09:35:42

Re: Perms and Combs! and some induction..

#72 2011-05-31 15:16:47

Re: Perms and Combs! and some induction..

#73 2011-05-31 16:25:49

Re: Perms and Combs! and some induction..

#74 2011-05-31 16:37:04

Re: Perms and Combs! and some induction..

#75 2011-05-31 17:09:44

Re: Perms and Combs! and some induction..

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