Probability of 3 Married Couples Winning a Prize

[haoku]

Homework Statement

There are 8 opposite-sex married couples in a party and 6 people are chosen to win a prize.
What is the probability that they are 3 married couple?

Homework Equations

P=6!/C(16,6)

The Attempt at a Solution

I try to do the question as
P=6!/C(16,6)
because there are 6! way which how many ways that 6 person can be choose from and choosing 6 people from 16 people C(16,6)
I know this equation is incorrect, but I have no idea how to do this question.
Could you please help me?

Thanks

 

[Hurkyl]

Instead of trying to jump straight to the answer... try finding the first step.

 

[Hurkyl]

(Don't read this post until you've thought my last one over)







One thing I often do first is to simply rewrite the problem in a different form. If I let

P = the probability that they are 3 married couples

Q = the number of ways to choose 3 married couple
R = the number of ways to choose 6 people

then

P = Q / R.


(Aside: I wanted to point out that an unstated assumption of the problem is that the choices are made uniformly randomly; each possible way to choose 6 people is exactly as likely as any other way)

 

[haoku]

The first step should be?
Total possible outcome: C(16,6)
Favourable outcome?

 

[haoku]

(Don't read this post until you've thought my last one over)







One thing I often do first is to simply rewrite the problem in a different form. If I let

P = the probability that they are 3 married couples

Q = the number of ways to choose 3 married couple
R = the number of ways to choose 6 people

then

P = Q / R.


(Aside: I wanted to point out that an unstated assumption of the problem is that the choices are made uniformly randomly; each possible way to choose 6 people is exactly as likely as any other way)

The Q is what I stuck into

 

[haoku]

Is the Q =8*7*6 ?
That is number if ways which 3 couples chosen from 8 couples

 

[Hurkyl]

note: the text in red is incorrect

Almost: you wrote down the number of permutations, not the number of combinations.

You can (correctly) work the problem by computing
permutations of 3 couples from 8 / permutations of 6 people from 18[/color]
or by computing
combinations of 3 couples from 8 / combinations of 6 people from 18

but mixing the two is wrong.

Although either of these ways work, using combinations is a direct translation of the problem, and is probably the way you should think about it.

 

[haoku]

Is the probability is:

P= C(8,3)/C(16,6)

?

 

[Hurkyl]

How did you show that Q = C(6,3)? How did you show that R = C(16,6)?

 

[haoku]

Almost: you wrote down the number of permutations, not the number of combinations.

You can (correctly) work the problem by computing
permutations of 3 couples from 8 / permutations of 6 people from 18
or by computing
combinations of 3 couples from 8 / combinations of 6 people from 18

but mixing the two is wrong.

Although either of these ways work, using combinations is a direct translation of the problem, and is probably the way you should think about it.

On more question:
permutations of 3 couples from 8 / permutations of 6 people from 18
is not equal to combinations of 3 couples from 8 / combinations of 6 people from 18
?

 

[Hurkyl]

On more question:
permutations of 3 couples from 8 / permutations of 6 people from 18
is not equal to combinations of 3 couples from 8 / combinations of 6 people from 18
?

Those numbers are equal.

 

[haoku]

How did you show that Q = C(6,3)? How did you show that R = C(16,6)?

R =C(16,6) because that is the number of ways that 6 people can be chosen from 16 people, regardless the order.

Q=C(8,3) is the number of ways to choose 3 couple from 8 couple.
(I have correct it, C(6,3) to C(8,3))

I think I am still missing something, because the lucky draw choose person one by one, not couple by couple, I need to multiply something to C(8,3)

 

[Hurkyl]

R =C(16,6) because that is the number of ways that 6 people can be chosen from 16 people, regardless the order.

Q=C(8,3) is the number of ways to choose 3 couple from 8 couple.
(I have correct it, C(6,3) to C(8,3))

So if you are confident in these calculations, and you are confident that P=Q/R, then you should be confident your answer!

I think I am still missing something, because the lucky draw choose person one by one

In the problem you stated, the draw was simply to choose six. The method of choosing 6 people doesn't effect the number of combinations.


But if you want to work out the problem as if the order of drawing the people matters, then you can still do it. It would look something like

P = Q/R
Q = Number of permutations of 6 people from 16, such that the 6 form 3 married couple
R = Number of permutations of 6 people from 16

and then you'd split Q up into two pieces, maybe

Q = S T
S = Number of ways to choose 3 married couples from 8
T = Number of ways to arrange those 6 people


There are other ways you might effect this calculation: e.g.

Q = AB + CD + EF
A = < Number of ways to choose 1 couple from 8 >
B = < Number of ways to place those 2 people into 6 slots >
C = < Number of ways to choose 1 couple from the remaining 7 >
D = < Number of ways to place those 2 people into the remaining 4 slots >
E = < Number of ways to choose 1 couple from the remaining 6 >
F = < Number of ways to place those 2 people into the remaining 2 slots >

(note: I think this Q overcounts by a factor of 6. I'm too tired to work it out)

 

[haoku]

Those numbers are equal.

When looking at definition, C(a,b)/C(c,d) =[ a!d!(c-d)!]/[c!b!(a-b)!]
P(a,b)/P(c,d)=[a!(c-d)!]/[c!(a-b)!]

So, the two number is not equal.

 

[Hurkyl]

You're right, silly mistake on my part to say that. It's not the permutations of the couples that goes on the numerator, but instead the permutations of the people in the couples.

(p.s. I've made an addition to my previous post)

 

[haoku]

So if you are confident in these calculations, and you are confident that P=Q/R, then you should be confident your answer!



In the problem you stated, the draw was simply to choose six. The method of choosing 6 people doesn't effect the number of combinations.


But if you want to work out the problem as if the order of drawing the people matters, then you can still do it. It would look something like

P = Q/R
Q = Number of permutations of 6 people from 16, such that the 6 form 3 married couple
R = Number of permutations of 6 people from 16

and then you'd split Q up into two pieces, maybe

Q = S T
S = Number of ways to choose 3 married couples from 8
T = Number of ways to arrange those 6 people


There are other ways you might effect this calculation: e.g.

Q = AB + CD + EF
A = < Number of ways to choose 1 couple from 8 >
B = < Number of ways to place those 2 people into 6 slots >
C = < Number of ways to choose 1 couple from the remaining 7 >
D = < Number of ways to place those 2 people into the remaining 4 slots >
E = < Number of ways to choose 1 couple from the remaining 6 >
F = < Number of ways to place those 2 people into the remaining 2 slots >

Oh yes, I have some logic error that confuse myself. Thanks

 

[Hurkyl]

Q = AB + CD + EF
A = < Number of ways to choose 1 couple from 8 >
B = < Number of ways to place those 2 people into 6 slots >
C = < Number of ways to choose 1 couple from the remaining 7 >
D = < Number of ways to place those 2 people into the remaining 4 slots >
E = < Number of ways to choose 1 couple from the remaining 6 >
F = < Number of ways to place those 2 people into the remaining 2 slots >

Argh, I've overcounted by a factor of 6, I think. I shouldn't do probability this late at night!