this probability question arised from the latest episode of TV show Survivor, a show which I am sure everyone knows about. The latest episode can been seen here within US: https://www.cbs.com/shows/survivor/video/pCyKg5imLvLv6Bb3N_DwV_3KQL4BAMrN/survivor-ghost-island-fate-is-the-homie/

In the episode there were two original tribes (tribe = group of people playing together): Orange with 6 players and Purple with 9 players. These 15 players were randomly divided into three new tribes: A, B and C so that each tribe consisted of 5 members. The draw was conducted by contestants blindly picking buffs in turn, there were 5 buffs of each kind A,B and C.

What happened in the draw was that the Orange minority tribe with 6 members ended up being split 2-2-2 into the three new tribes, meaning that each tribe A,B,C ended up with 2 Orange and 3 Purple contestants. Can anyone calculate the probability of this outcome, because an uneven split seems to be much more likely?

To put the question into simpler from: You can think of there being 9 Purple balls and 6 Orange balls. The balls are randomly labeled A,B,C so that there are 5 of each A,B,C. What are the odds that exactly 2 Orange balls get label A, B and C?

Quote:Jufo81...What happened in the draw was that the Orange minority tribe with 6 members ended up being split 2-2-2 into the three new tribes, meaning that each tribe A,B,C ended up with 2 Orange and 3 Purple contestants. Can anyone calculate the probability of this outcome, because an uneven split seems to be much more likely?...

I get 0.12397 as the probability. And I find that 2 Orange with 3 Purple in each new tribe is the most likely outcome. I, too, expected an uneven distribution more likely.

By the way, the number of ways for 2 Oranges in each of the first two tribes is combin(6,2) * combin(4,2) = 90.

There are (15)C(6) (i.e. COMBIN(15,6)) = 5005 ways to draw 6 balls from the 15. There are (5)C(2) = 10 pairs of As that can be drawn, (5)C(2) pairs of Bs, and (5)C(2) pairs of Cs, so the probability = (10 x 10 x 10) / 5005 = 200 / 1001.

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The chances that any one of these combinations is drawn for the first group is 10[(6*5*9*8*7)/(15*14*13*12*11)]=60/143=0.41958

Given that the first group met the requirements, that chances that any one of the combination is drawn for the second group is 10[(4*3*6*5*4)/(10*9*8*7*6)]=10/21=0.47619

The third group, by default has to meet the requirements.

Therefore, multiplying these two probabilities above gives us 200/1001= 19.98%

Quote:MidwestAPI get the same result at ThatDonGuy. There are 10 ways that exactly two orange end up in any group.

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The chances that any one of these combinations is drawn for the first group is 10[(6*5*9*8*7)/(15*14*13*12*11)]=60/143=0.41958

Given that the first group met the requirements, that chances that any one of the combination is drawn for the second group is 10[(4*3*6*5*4)/(10*9*8*7*6)]=10/21=0.47619

The third group, by default has to meet the requirements.

Therefore, multiplying these two probabilities above gives us 200/1001= 19.98%

Thanks for the explanation. I now agree with you and Don.

Quote:MidwestAPTherefore, multiplying these two probabilities above gives us 200/1001= 19.98%

I agree. Here is my math: =(COMBIN(6,2)*COMBIN(9,3)/COMBIN(15,5))*(COMBIN(4,2)*COMBIN(6,3)/COMBIN(10,5))

Where combin(x,y)=x!/((y!*(x-y)!)

Where x! = 1*2*3*...*x

I haven't seen the episode yet. I tend to lose interest as the large-breasted women get voted out, which was the case last week.

Quote:ChesterDog

I get 0.12397 as the probability. And I find that 2 Orange with 3 Purple in each new tribe is the most likely outcome. I, too, expected an uneven distribution more likely.

By the way, the number of ways for 2 Oranges in each of the first two tribes is combin(6,2) * combin(4,2) = 90.

Hmm I got the same result as ChesterDog: 0.12397. All the possible ways to split the 6 Orange balls into three groups of five are {5,1,0},{4,2,0},{4,1,1},{3,2,1},{3,3,0},{2,2,2}, and out of these {3,2,1} is by far the most likely, not {2,2,2}.

If we start with {5,1,0} it can be ordered in 6 ways ({5,1,0},{5,0,1},{1,0,5},{1,5,0},{0,1,5},{0,5,1}) each of which is equally likely. If we label the orange balls a...f there are six ways to put them in {5,1,0} so the total number of these combinations is 6*6 = 36.

Similarily the total number of combinations for the other cases:

{4,2,0}: 6*(6)C(2) = 90

{4,1,1}: 3*6*5 = 90

{3,2,1}: 6*6*(5)C(2) = 360

{3,3,0}: 3*(6)C(3) = 60

{2,2,2}: (6)C(2)*(4)C(2) = 90 (<- what ChesterDog also wrote)

So in total there are 36 + 90 + 90 + 360 + 60 + 90 = 726 combinations and thus the probability for {2,2,2} split is 90/726 = 0.12397.

If this is wrong and the answer is 19.98% why is this so?

Quote:Jufo81So in total there are 36 + 90 + 90 + 360 + 60 + 90 = 726 combinations and thus the probability for {2,2,2} split is 90/726 = 0.12397.

If this is wrong and the answer is 19.98% why is this so?

You are not counting the number of ways the 9 purple balls can be put into the remaining positions.

For example, each of the 36 {5,1,0} distributions of orange balls has (9)C(4) = 126 {0,1,5} distributions of purple balls.

For each distribution of orange balls, the total number of (orange ball distributions) x (purple ball distributions) is:

5,1,0: 36 x 126 = 4536

4,2,0: 90 x 504 = 45360

4,1,1: 90 x 630 = 56700

3,2,1: 360 x 1260 = 453600

3,3,0: 60 x 756 = 45360

2,2,2: 90 x 1680 = 151200

The ratio of 2,2,2 distributions to the total = 151,200 / 756,756 = 200 / 1001.

Also notice that 3,2,1 is the most likely distribution.

Quote:ThatDonGuyYou are not counting the number of ways the 9 purple balls can be put into the remaining positions.

For example, each of the 36 {5,1,0} distributions of orange balls has (9)C(4) = 126 {0,1,5} distributions of purple balls.

For each distribution of orange balls, the total number of (orange ball distributions) x (purple ball distributions) is:

5,1,0: 36 x 126 = 4536

4,2,0: 90 x 504 = 45360

4,1,1: 90 x 630 = 56700

3,2,1: 360 x 1260 = 453600

3,3,0: 60 x 756 = 45360

2,2,2: 90 x 1680 = 151200

The ratio of 2,2,2 distributions to the total = 151,200 / 756,756 = 200 / 1001.

Also notice that 3,2,1 is the most likely distribution.

Thanks for this, I get it now!