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Six 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1 and exactly two of the dice show a 2? Express your answer as a common fraction.

 Dec 31, 2019
 #1
avatar
+2

My short computer code indicates that the total should be = 1440.
Now, here is my attempt to justify the above number mathematically!
We have 6 C 4 =15 ways of chosing the 1s and 2s.
We have 4 C 2 =6 ways of chosing the remaining 2 numbers.
The remaining 2 numbers (3, 4, 5, 6) =4 can be selected in 4 x 4 = 16 ways.
So, the overall total should be: 15 x 6 x 16 =1440 ways, which is in agreement with my computer code.

Therefore the probability should be: 1440 / 6^6 = 5 / 162.
Note: Some mathematician should check this!!.

 Dec 31, 2019
 #2
avatar+109509 
+1

 

Six 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1 and exactly two of the dice show a 2? Express your answer as a common fraction.

 

My answer does not agree with yours guest.  Generally I have a lot of faith in the output of people's computer programs.  

I certainly do not have full confidence in my answer but the logic you have used to justify your programmng output does not make sense to me.

I think we need another adjudicator.  indecision

Can I see you code please. It probably will not make sense to me, my programming skills are way our of date, but you never know maybe I wills see somthing there.  It is a good idea to include your code for these answers anyway.

 

Here is my logic:

 

1,1,2,2,x,y,

possibilities

 

x and y are both the same.     4*6!/2!2!2! = 4*90 = 360  ways

 

x and y are different.             4*3*6!/2!2! = 12*180 =   2160 ways

 

Total of 360+2160 = 2520  ways

 

Sample space = 6^6=46656

 

Prob = 2520/46656 = 35/648 = approx    0.054

 Dec 31, 2019
 #3
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0

.......

 Dec 31, 2019
edited by Guest  Dec 31, 2019
 #4
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+1

Hi Melody: Here is a short computer code which is very brief version of the full code. It calculates 16 combinations for for 1 of 6 numbers. To write a "full conditional statement", you have to repeat the "if condition" 8 x 6 x 2 =96 times !!!! So, I simply took a "shortcut" and arrived at 1440 as explained below the code:

 

a=0;b=1;c=1;d=1;e=1;f=1;g=1;p=0; cycle:n=b*100000+c*10000+d*1000+e*100+f*10+g; if(b==1 and c==1 and d==2 and e==2  and f!=1 and f!=2 and g!=1 and g!=2, goto loop, goto next);loop:printn," ",;p=p+1; next:g++;if(g<7, goto cycle, 0);g=1;f++;if(f<7, goto cycle, 0);g=1;f=1;e++;if(e<7, goto cycle,0);g=1;f=1;e=1;d++;if(d<7, goto cycle,0);g=1;f=1;e=1;d=1;c++;if(c<7, goto cycle,0);g=1;f=1;e=1;d=1;c=1;b++;if(b<7, goto cycle,0);g=1;f=1;e=1;d=1;c=1;b=1;a++;if(a==0, goto cycle,0);print"Total = ",p

 

OUTPUT: 112233  112234  112235  112236  112243  112244  112245  112246  112253  112254  112255  112256  112263  112264  112265  112266  Total =  16
This is only for 1 number. Then you cycle throug 6 numbers =16 x 6 = 96. and this is only one go for 15 combinations:6 C 4 = 15. And 15 x 96 =1440. This is how I arrived at 1440.

 

P.S. I found your mistake in this calculation:   x and y are different. 4*3*6!/2!2! = 12*180 =   2160 ways

The mistake is : when x, y are different, you only have 6 choices not 12 as follows: (3,4), (3,5), (3,6), (4,5), (4,6), (5,6) making your calculation =6 * 6! / 2!2! =1080. Then add it to your first calculation and we get:

1080 + 360 = 1440 which agrees with the computer output.

 Dec 31, 2019
edited by Guest  Dec 31, 2019
 #6
avatar+109509 
+2

Thanks for all that information, I do appreciate your efforts.

 

But I do not think that is correct.

The way i have done it, 

Is to think of every dice as being distinguishable.  Like they are all different colours so

if x and y are different (3,4) is different from (4,3)

 

So what I think is,

You have found how many combinations there are that satisfy the conditions given that the dice are indistinguishable.

then

You have divided by by the total number of ways WHEN the dice are different from each other, (distinguishable)

 

So I think you have mixed identical dice in the numberator with different looking dice in the denominator.

 

So I still think my answer is more likely to be correct than your is.

Melody  Dec 31, 2019
 #8
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+2

Hi Melody: Bravo!!! You were right after all. I simply computed the permutations of the (3,4,5,6) =4 x 4 =16 and presto got the same number that you did. I believe my mistake was in using 6 C 4, which I believe now to be wrong, because 4 are 2 separate entities(1,1 and 2,2).
112233    = ........................................................90 permutations  
112234  112243    =..........................................360p
112235  112244  112253 =...............................450p
112236  112245  112254  112263=..................720p  
112246  112255  112264= ...............................450p  
112256  112265=..............................................360p  
112266=..............................................................90p
                                                GRAND TOTAL =2,520 PERMUTATIONS.  
 

Guest Jan 1, 2020
 #9
avatar+109509 
+2

Thanks Guest,

I am glad that we are now in agreement. That is always comforting.     laugh

Melody  Jan 1, 2020
 #17
avatar+1904 
+2

You should know, Mr. BB, that using “brovos” for a woman is often considered mockery. The correct form is “brave” and “brava” is the singular form.

In any case, Melody's brave should be directed here, https://web2.0calc.com/questions/six-6-sided-dice-are-rolled-what-is-the-probability, where Melody and I monkey around with this exact question.

 

 

GA

GingerAle  Jan 2, 2020
 #18
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0

 bravo!

Guest Jan 5, 2020
 #5
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+1

Sorry Melody: I thought I had found a mistake, which on second thought, is NOT a mistake !!!. I was taking "combinations of the 4 numbers, instead of their permutation", which you have it accurately counted. Shows you how much I know about combinations and permutations! I don't really know where the difference of 2520 - 1440 =1080 is !!.

 

By the way, I was able to list 1/3 of 1440 = 480 permutations, and here is the list. See if you can find any mistakes or duplicates in it:

 

(112233, 112234, 112235, 112236, 112243, 112244, 112245, 112246, 112253, 112254, 112255, 112256, 112263, 112264, 112265, 112266, 112332, 112342, 112352, 112362, 112432, 112442, 112452, 112462, 112532, 112542, 112552, 112562, 112632, 112642, 112652, 112662, 113223, 113224, 113225, 113226, 113232, 113242, 113252, 113262, 113322, 113422, 113522, 113622, 114223, 114224, 114225, 114226, 114232, 114242, 114252, 114262, 114322, 114422, 114522, 114622, 115223, 115224, 115225, 115226, 115232, 115242, 115252, 115262, 115322, 115422, 115522, 115622, 116223, 116224, 116225, 116226, 116232, 116242, 116252, 116262, 116322, 116422, 116522, 116622, 121233, 121234, 121235, 121236, 121243, 121244, 121245, 121246, 121253, 121254, 121255, 121256, 121263, 121264, 121265, 121266, 121332, 121342, 121352, 121362, 121432, 121442, 121452, 121462, 121532, 121542, 121552, 121562, 121632, 121642, 121652, 121662, 122133, 122134, 122135, 122136, 122143, 122144, 122145, 122146, 122153, 122154, 122155, 122156, 122163, 122164, 122165, 122166, 122331, 122341, 122351, 122361, 122431, 122441, 122451, 122461, 122531, 122541, 122551, 122561, 122631, 122641, 122651, 122661, 123213, 123214, 123215, 123216, 123321, 123421, 123521, 123621, 124213, 124214, 124215, 124216, 124321, 124421, 124521, 124621, 125213, 125214, 125215, 125216, 125321, 125421, 125521, 125621, 126213, 126214, 126215, 126216, 126321, 126421, 126521, 126621, 131232, 131242, 131252, 131262, 131322, 131422, 131522, 131622, 132132, 132142, 132152, 132162, 133212, 134212, 135212, 136212, 141232, 141242, 141252, 141262, 141322, 141422, 141522, 141622, 142132, 142142, 142152, 142162, 143212, 144212, 145212, 146212, 151232, 151242, 151252, 151262, 151322, 151422, 151522, 151622, 152132, 152142, 152152, 152162, 153212, 154212, 155212, 156212, 161232, 161242, 161252, 161262, 161322, 161422, 161522, 161622, 162132, 162142, 162152, 162162, 163212, 164212, 165212, 166212, 211233, 211234, 211235, 211236, 211243, 211244, 211245, 211246, 211253, 211254, 211255, 211256, 211263, 211264, 211265, 211266, 211332, 211342, 211352, 211362, 211432, 211442, 211452, 211462, 211532, 211542, 211552, 211562, 211632, 211642, 211652, 211662, 212133, 212134, 212135, 212136, 212143, 212144, 212145, 212146, 212153, 212154, 212155, 212156, 212163, 212164, 212165, 212166, 212331, 212341, 212351, 212361, 212431, 212441, 212451, 212461, 212531, 212541, 212551, 212561, 212631, 212641, 212651, 212661, 213123, 213124, 213125, 213126, 213312, 213412, 213512, 213612, 214123, 214124, 214125, 214126, 214312, 214412, 214512, 214612, 215123, 215124, 215125, 215126, 215312, 215412, 215512, 215612, 216123, 216124, 216125, 216126, 216312, 216412, 216512, 216612, 221133, 221134, 221135, 221136, 221143, 221144, 221145, 221146, 221153, 221154, 221155, 221156, 221163, 221164, 221165, 221166, 221331, 221341, 221351, 221361, 221431, 221441, 221451, 221461, 221531, 221541, 221551, 221561, 221631, 221641, 221651, 221661, 223113, 223114, 223115, 223116, 223131, 223141, 223151, 223161, 223311, 223411, 223511, 223611, 224113, 224114, 224115, 224116, 224131, 224141, 224151, 224161, 224311, 224411, 224511, 224611, 225113, 225114, 225115, 225116, 225131, 225141, 225151, 225161, 225311, 225411, 225511, 225611, 226113, 226114, 226115, 226116, 226131, 226141, 226151, 226161, 226311, 226411, 226511, 226611, 231231, 231241, 231251, 231261, 232131, 232141, 232151, 232161, 232311, 232411, 232511, 232611, 233121, 234121, 235121, 236121, 241231, 241241, 241251, 241261, 242131, 242141, 242151, 242161, 242311, 242411, 242511, 242611, 243121, 244121, 245121, 246121, 251231, 251241, 251251, 251261, 252131, 252141, 252151, 252161, 252311, 252411, 252511, 252611, 253121, 254121, 255121, 256121, 261231, 261241, 261251, 261261, 262131, 262141, 262151, 262161, 262311, 262411, 262511, 262611, 263121, 264121, 265121, 266121) Total = 480
 

 Dec 31, 2019
edited by Guest  Dec 31, 2019
 #10
avatar
0

Guest's first answer is correct

 Jan 1, 2020
 #11
avatar+30038 
+5

Here is a Monte Carlo program (in MathCad) that supports Guest #1's result:

 

Of course, since the Monte Carlo program depends on random numbers the result it returns may be slightly different each time it runs.  However, it does seem to provide strong support for Guest #1.

 Jan 1, 2020
 #12
avatar
+1

Hello Alan: That conclusion is extremely surprising in light of having just completed the individual permutations of each of the following 16 groups. Because I permuted each of the 16 groups separately, I don't think the possibility of duplication is likely. Finally, I assembled the entire list and the computer counted them as the total of these 16 individual permutations and both agreed as to the total of =2,520 permutations!
So, Alan, where could the overcounting possibly be? Any ideas where to look for possible duplications?
1 - 112233 = 90 permutations
2 - 112234 = 180p 
3 - 112235 = 180p 
4 - 112236 = 180p 
5 - 112243 = 180p 
6 - 112244 = 90p 
7 - 112245 = 180p 
8 - 112246 = 180p 
9 - 112253 = 180p 
10 -112254 = 180p 
11 -112255 = 90p 
12 -112256 = 180p
13 -112263 = 180p 
14 -112264 = 180p 
15 -112265 = 180p 
16 -112266 = 90p
Here is a complete list of all 90 permutations of the first group:112233. See if you can find anything wrong with it. Thanks.
{1, 1, 2, 2, 3, 3}, {1, 1, 2, 3, 2, 3}, {1, 1, 2, 3, 3, 2}, {1, 1, 3, 2, 2, 3}, {1, 1, 3, 2, 3, 2}, {1, 1, 3, 3, 2, 2}, {1, 2, 1, 2, 3, 3}, {1, 2, 1, 3, 2, 3}, {1, 2, 1, 3, 3, 2}, {1, 2, 2, 1, 3, 3}, {1, 2, 2, 3, 1, 3}, {1, 2, 2, 3, 3, 1}, {1, 2, 3, 1, 2, 3}, {1, 2, 3, 1, 3, 2}, {1, 2, 3, 2, 1, 3}, {1, 2, 3, 2, 3, 1}, {1, 2, 3, 3, 1, 2}, {1, 2, 3, 3, 2, 1}, {1, 3, 1, 2, 2, 3}, {1, 3, 1, 2, 3, 2}, {1, 3, 1, 3, 2, 2}, {1, 3, 2, 1, 2, 3}, {1, 3, 2, 1, 3, 2}, {1, 3, 2, 2, 1, 3}, {1, 3, 2, 2, 3, 1}, {1, 3, 2, 3, 1, 2}, {1, 3, 2, 3, 2, 1}, {1, 3, 3, 1, 2, 2}, {1, 3, 3, 2, 1, 2}, {1, 3, 3, 2, 2, 1}, {2, 1, 1, 2, 3, 3}, {2, 1, 1, 3, 2, 3}, {2, 1, 1, 3, 3, 2}, {2, 1, 2, 1, 3, 3}, {2, 1, 2, 3, 1, 3}, {2, 1, 2, 3, 3, 1}, {2, 1, 3, 1, 2, 3}, {2, 1, 3, 1, 3, 2}, {2, 1, 3, 2, 1, 3}, {2, 1, 3, 2, 3, 1}, {2, 1, 3, 3, 1, 2}, {2, 1, 3, 3, 2, 1}, {2, 2, 1, 1, 3, 3}, {2, 2, 1, 3, 1, 3}, {2, 2, 1, 3, 3, 1}, {2, 2, 3, 1, 1, 3}, {2, 2, 3, 1, 3, 1}, {2, 2, 3, 3, 1, 1}, {2, 3, 1, 1, 2, 3}, {2, 3, 1, 1, 3, 2}, {2, 3, 1, 2, 1, 3}, {2, 3, 1, 2, 3, 1}, {2, 3, 1, 3, 1, 2}, {2, 3, 1, 3, 2, 1}, {2, 3, 2, 1, 1, 3}, {2, 3, 2, 1, 3, 1}, {2, 3, 2, 3, 1, 1}, {2, 3, 3, 1, 1, 2}, {2, 3, 3, 1, 2, 1}, {2, 3, 3, 2, 1, 1}, {3, 1, 1, 2, 2, 3}, {3, 1, 1, 2, 3, 2}, {3, 1, 1, 3, 2, 2}, {3, 1, 2, 1, 2, 3}, {3, 1, 2, 1, 3, 2}, {3, 1, 2, 2, 1, 3}, {3, 1, 2, 2, 3, 1}, {3, 1, 2, 3, 1, 2}, {3, 1, 2, 3, 2, 1}, {3, 1, 3, 1, 2, 2}, {3, 1, 3, 2, 1, 2}, {3, 1, 3, 2, 2, 1}, {3, 2, 1, 1, 2, 3}, {3, 2, 1, 1, 3, 2}, {3, 2, 1, 2, 1, 3}, {3, 2, 1, 2, 3, 1}, {3, 2, 1, 3, 1, 2}, {3, 2, 1, 3, 2, 1}, {3, 2, 2, 1, 1, 3}, {3, 2, 2, 1, 3, 1}, {3, 2, 2, 3, 1, 1}, {3, 2, 3, 1, 1, 2}, {3, 2, 3, 1, 2, 1}, {3, 2, 3, 2, 1, 1}, {3, 3, 1, 1, 2, 2}, {3, 3, 1, 2, 1, 2}, {3, 3, 1, 2, 2, 1}, {3, 3, 2, 1, 1, 2}, {3, 3, 2, 1, 2, 1}, {3, 3, 2, 2, 1, 1} = 90 permutations

 Jan 1, 2020
 #13
avatar+30038 
+4

I'm not sure how you've calculated your permutations, but it looks to me as though your groups 1122xy are the same as groups 1122yx ((where x and y are different).

Certainly, if you remove those repeats (if indeed they are repeats) you get back to the 1440/6^6 result.

Alan  Jan 1, 2020
edited by Alan  Jan 1, 2020
 #14
avatar
0

Thanks Alan: I ran them on WolframAlpha and, in my experience, it NEVER lists duplicates! Please be good enough to look at this Wolfram page where I ran the 180 permutations of:112234:

 

https://www.wolframalpha.com/input/?i=permutations+of+%7B1%2C+1%2C+2%2C+2%2C+3%2C+4%7D

 

P.S. if you find  it difficult to read, you may press the "plain text" button, which might be easier to read. Thanks.

 Jan 1, 2020
 #15
avatar+30038 
+4

Yes, but if you look at the set of permutations for 112234, say, in WolframAlpha, you will see that it is identical to that for 112243.

Alan  Jan 1, 2020
 #16
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0

Thanks Alan. You are absolutely right. I just found out this about Wolfram: when your enter a set of numbers such as:112243, the first thing the computing engine does is to put them in "numerical order" and then calculate the permutations on them in numerical order. in other words, it treats:112234 exactly the same as:112243, hence the duplication in reversed groups of: 34 and 43, 35 and 53, 36 and 63, 45 and 54, 46 and 64, 56 and 65.
And that is the difference of 1,080 =6 groups x 180

 Jan 1, 2020
 #19
avatar+109509 
0

Thanks Alan and guest.  

Some time maybe I will work out why mine is wrong.

It is always good to have the correct answer anyway.  

 

What would we do without your Monty Carlo simulations.  They really ae very useful !

 Jan 5, 2020
 #20
avatar+117 
+1

I arrived at the now accepted result, but via a different method.

Think of a qualifying sequence as any containing exactly two ones,two twos, and two "others".

Every qualifying sequence has the same probability of occurring, (1/6).(1/6).(1/6).(1/6).(4/6).(4/6)=16/(6^6).

The order doesn't matter.

The number of qualifying sequences will be 6!/(2^3), the three twos to remove the duplication of the 1's, the 2's, and the "others".

The total probability will be the product of these. 

That works out to be 5/162, approximately 0.030864.

 Jan 5, 2020

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