Answer:
(a) The probability that the members of the committee are chosen from all nationalities
=0.1212.
(b)The probability that all nationalities except Italian are represent is 0.04848.
Step-by-step explanation:
Hypergeometric Distribution:
Let
,
,
and
be four given positive integers and let
.
A random variable X is said to have hypergeometric distribution with parameter
,
,
,
and n.
The probability mass function
![f(x_1,x_2.x_3,x_4;a_1,a_2,a_3,a_4;N,n)=\frac{\left(\begin{array}{c}x_1\\a_1\end{array}\right)\left(\begin{array}{c}x_2\\a_2\end{array}\right) \left(\begin{array}{c}x_3\\a_3\end{array}\right) \left(\begin{array}{c}x_4\\a_4\end{array}\right) }{\left(\begin{array}{c}N\\n\end{array}\right) }](https://tex.z-dn.net/?f=f%28x_1%2Cx_2.x_3%2Cx_4%3Ba_1%2Ca_2%2Ca_3%2Ca_4%3BN%2Cn%29%3D%5Cfrac%7B%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7Dx_1%5C%5Ca_1%5Cend%7Barray%7D%5Cright%29%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7Dx_2%5C%5Ca_2%5Cend%7Barray%7D%5Cright%29%20%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7Dx_3%5C%5Ca_3%5Cend%7Barray%7D%5Cright%29%20%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7Dx_4%5C%5Ca_4%5Cend%7Barray%7D%5Cright%29%20%20%7D%7B%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7DN%5C%5Cn%5Cend%7Barray%7D%5Cright%29%20%7D)
Here ![a_1+a_2+a_3+a_4=n](https://tex.z-dn.net/?f=a_1%2Ba_2%2Ba_3%2Ba_4%3Dn)
![{\left(\begin{array}{c}x_1\\a_1\end{array}\right)=^{x_1}C_{a_1}= \frac{x_1!}{a_1!(x_1-a_1)!}](https://tex.z-dn.net/?f=%7B%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7Dx_1%5C%5Ca_1%5Cend%7Barray%7D%5Cright%29%3D%5E%7Bx_1%7DC_%7Ba_1%7D%3D%20%5Cfrac%7Bx_1%21%7D%7Ba_1%21%28x_1-a_1%29%21%7D)
Given that, a foreign club is made of 2 Canadian members, 3 Japanese members, 5 Italian members and 2 Germans members.
=2,
=3,
=5 and
=2.
A committee is made of 4 member.
N=4
(a)
We need to find out the probability that the members of the committee are chosen from all nationalities.
=1,
=1,
=1 ,
=1, n=4
The required probability is
![=\frac{\left(\begin{array}{c}2\\1\end{array}\right)\left(\begin{array}{c}3\\1\end{array}\right) \left(\begin{array}{c}5\\1\end{array}\right) \left(\begin{array}{c}2\\1\end{array}\right) }{\left(\begin{array}{c}12\\4\end{array}\right) }](https://tex.z-dn.net/?f=%3D%5Cfrac%7B%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D2%5C%5C1%5Cend%7Barray%7D%5Cright%29%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D3%5C%5C1%5Cend%7Barray%7D%5Cright%29%20%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D5%5C%5C1%5Cend%7Barray%7D%5Cright%29%20%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D2%5C%5C1%5Cend%7Barray%7D%5Cright%29%20%20%7D%7B%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D12%5C%5C4%5Cend%7Barray%7D%5Cright%29%20%7D)
![=\frac{2\times 3\times 5\times 2}{495}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B2%5Ctimes%203%5Ctimes%205%5Ctimes%202%7D%7B495%7D)
![=\frac{4}{33}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B4%7D%7B33%7D)
=0.1212
(b)
Now we find out the probability that all nationalities except Italian.
So, we need to find out,
![P(a_1=2,a_2=1,a_3=0,a_4=1)+P(a_1=1,a_2=2,a_3=0,a_4=1)+P(a_1=1,a_2=1,a_3=0,a_4=2)](https://tex.z-dn.net/?f=P%28a_1%3D2%2Ca_2%3D1%2Ca_3%3D0%2Ca_4%3D1%29%2BP%28a_1%3D1%2Ca_2%3D2%2Ca_3%3D0%2Ca_4%3D1%29%2BP%28a_1%3D1%2Ca_2%3D1%2Ca_3%3D0%2Ca_4%3D2%29)
![=\frac{\left(\begin{array}{c}2\\2\end{array}\right)\left(\begin{array}{c}3\\1\end{array}\right) \left(\begin{array}{c}5\\0\end{array}\right) \left(\begin{array}{c}2\\1\end{array}\right) }{\left(\begin{array}{c}12\\4\end{array}\right) }+\frac{\left(\begin{array}{c}2\\1\end{array}\right)\left(\begin{array}{c}3\\2\end{array}\right) \left(\begin{array}{c}5\\0\end{array}\right) \left(\begin{array}{c}2\\1\end{array}\right) }{\left(\begin{array}{c}12\\4\end{array}\right) }](https://tex.z-dn.net/?f=%3D%5Cfrac%7B%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D2%5C%5C2%5Cend%7Barray%7D%5Cright%29%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D3%5C%5C1%5Cend%7Barray%7D%5Cright%29%20%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D5%5C%5C0%5Cend%7Barray%7D%5Cright%29%20%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D2%5C%5C1%5Cend%7Barray%7D%5Cright%29%20%20%7D%7B%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D12%5C%5C4%5Cend%7Barray%7D%5Cright%29%20%7D%2B%5Cfrac%7B%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D2%5C%5C1%5Cend%7Barray%7D%5Cright%29%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D3%5C%5C2%5Cend%7Barray%7D%5Cright%29%20%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D5%5C%5C0%5Cend%7Barray%7D%5Cright%29%20%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D2%5C%5C1%5Cend%7Barray%7D%5Cright%29%20%20%7D%7B%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D12%5C%5C4%5Cend%7Barray%7D%5Cright%29%20%7D)
![+\frac{\left(\begin{array}{c}2\\1\end{array}\right)\left(\begin{array}{c}3\\1\end{array}\right) \left(\begin{array}{c}5\\0\end{array}\right) \left(\begin{array}{c}2\\2\end{array}\right) }{\left(\begin{array}{c}12\\4\end{array}\right) }](https://tex.z-dn.net/?f=%2B%5Cfrac%7B%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D2%5C%5C1%5Cend%7Barray%7D%5Cright%29%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D3%5C%5C1%5Cend%7Barray%7D%5Cright%29%20%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D5%5C%5C0%5Cend%7Barray%7D%5Cright%29%20%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D2%5C%5C2%5Cend%7Barray%7D%5Cright%29%20%20%7D%7B%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D12%5C%5C4%5Cend%7Barray%7D%5Cright%29%20%7D)
![=\frac{1\times 3\times 1\times 2}{495}+\frac{2\times 3\times 1\times 2}{495}+\frac{2\times 3\times 1\times 1}{495}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%5Ctimes%203%5Ctimes%201%5Ctimes%202%7D%7B495%7D%2B%5Cfrac%7B2%5Ctimes%203%5Ctimes%201%5Ctimes%202%7D%7B495%7D%2B%5Cfrac%7B2%5Ctimes%203%5Ctimes%201%5Ctimes%201%7D%7B495%7D)
![=\frac{6+12+6}{495}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B6%2B12%2B6%7D%7B495%7D)
![=\frac{8}{165}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B8%7D%7B165%7D)
=0.04848
The probability that all nationalities except Italian are represent is 0.04848.