Question

From a group of 3 Indians, 4 Pakistanis and 5 Americans sub- committee are formed with 4 persons selected randomly. Find the probability that sub-committee are consist of i)2 Indians and 2 Pakistani ii) 1 Indian ,1pakistani and 2 Americans (iii) 1 Pakistani and 3 Americans​

Answer 1

Answer:

i) P (Sub-committee of 2 Indians and 2 Pakistanis) = 0.036

ii) P (Sub-committee of 1 Indian, 1 Pakistani and 2 Americans) = 0.24

iii) P (Sub-committee of 1 Pakistani and 3 Americans) = 0.889

Explanation:

Given:

No. of Indians = 3

No. of Pakistanis = 4

No. of Americans = 5

To find:

i) Probability of a subcommittee of 2 Indians and 2 Pakistanis

ii) Probability of a subcommittee of 1 Indian, 1 Pakistani and 2 Americans

iii) Probability of a subcommittee of 1 Pakistani and 3 Americans

Solution:

Total no. of people => 3 + 4 + 5 = 12

i) From a group of 3 Indians, 2 Indians can be selected 3C2 ways and from a group of 4 Pakistanis, 2 Pakistanis can be selected in 4C2 ways.

Total no. of ways in which 4 people can be selected from a group of 12 people = 12C4

Therefore, P (Sub-committee of 2 Indians and 2 Pakistanis) = No. of favourable selections/total no. of of possible selections

= (3C2 x 4C2) / 12C4

We know that, nCr = n! / r! x (n - r)!

Therefore, P (Sub-committee of 2 Indians and 2 Pakistanis) = 18/495

P (Sub-committee of 2 Indians and 2 Pakistanis) = 0.036

ii) From a group of 3 Indians, 1 Indian can be selected 3C1 ways and from a group of 4 Pakistanis, 1 Pakistani can be selected in 4C1 ways. From a group of 5 Americans, 2 can be selected in 5C2 ways.

Total no. of ways in which 4 people can be selected from a group of 12 people = 12C4

Therefore, P (Sub-committee of 1 Indian, 1 Pakistani and 2 Americans) = No. of favourable selections/total no. of of possible selections

= (3C1 x 4C1 x 5C2) / 12C4

We know that, nCr = n! / r! x (n - r)!

Therefore, P (Sub-committee of 1 Indian, 1 Pakistani and 2 Americans) = 120/495

P (Sub-committee of 1 Indian, 1 Pakistani and 2 Americans) = 0.24

iii) From a group of 4 Pakistanis, 1 Pakistani can be selected in 4C1 ways. From a group of 5 Americans, 3 can be selected in 5C3 ways.

Total no. of ways in which 4 people can be selected from a group of 12 people = 12C4

Therefore, P (Sub-committee of 1 Pakistani and 3 Americans) = No. of favourable selections/total no. of of possible selections

= (4C1 x 5C3) / 12C4

We know that, nCr = n! / r! x (n - r)!

Therefore, P (Sub-committee of 1 Pakistani and 3 Americans) = 40/495

P (Sub-committee of 1 Pakistani and 3 Americans) = 0.889

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Answer 2

Answer: P (Subcommittee of 1 Pakistani and 3 Americans) = 0.889

Explanation:

Given:

Number of Indians = 3

Number of Pakistanis = 4

Number of Americans = 5

Find:

i) Probability of a sub-committee of 2 Indians and 2 Pakistanis

ii) Probability of a subcommittee consisting of 1 Indian, 1 Pakistani, and 2 Americans

iii) Probability of a subcommittee consisting of 1 Pakistani and 3 Americans

Solution:

Total number of people => 3 + 4 + 5 = 12

i) From a group of 3 Indians, 2 Indians can be selected in $^{3} C_{2}$ ways, and from a group of 4 Pakistanis, 2 Pakistanis can be selected in $^{4} C_{2}$ ways.

Total number of ways in which 4 people can be selected from a group of 12 people = $^{12} C_{4}$

Hence P (Sub-Committee of 2 Indians and 2 Pakistanis) = Number of Favorable Selections/Total Number. from the possible selections

= $\frac{^{3} C_{2} *^{4} C_{2}}{^{12} C_{4}}$

We know that $^{n} C_{r}$= $\frac{n!}{r!(n-r)!}$

Hence P (subcommittee of 2 Indians and 2 Pakistanis) = $\frac{18}{495}$

P (Subcommittee of 2 Indians and 2 Pakistanis) = 0.036

ii) From a group of 3 Indians, 1 Indian can be selected in $^{3} C_{1}$ways, and from a group of 4 Pakistanis, 1 Pakistani can be selected in $^{4} C_{1}$ ways. 2 ways $^{5} C_{2}$ can be chosen from a group of 5 Americans.

Total number of ways in which 4 people can be selected from a group of 12 people = $^{12} C_{4}$

Therefore P (Sub-committee of 1 Indian, 1 Pakistani, and 2 Americans) = number of favorable selections/total number. from the possible selections

= $\frac{^{3} C_{1}*^{5} C_{2}}{^{12} C_{4}}$

We know that $^{n} C_{r}$ = $\frac{n!}{r!(n-r)!}$

Therefore P (a subcommittee of 1 Indian, 1 Pakistani, and 2 Americans) = $\frac{120}{495}$

P (a subcommittee of 1 Indian, 1 Pakistani, and 2 Americans) = 0.24

iii) From a pool of 4 Pakistanis, 1 Pakistani can be selected in $^{4} C_{1}$ ways. From a group of 5 Americans, 3 ways of 5C3 can be chosen.

Total number of ways in which 4 people can be selected from a group of 12 people = $^{12} C_{4}$

Therefore P (Sub-Committee 1 Pakistani and 3 Americans) = number of favorable selections/total number. from the possible selections

= $\frac{^{4} C_{1}*^{5} C_{3}}{^{12} C_{4}}$

We know that nCr = $\frac{n!}{r!(n-r)!}$

Therefore P (a subcommittee of 1 Pakistani and 3 Americans) = $\frac{40}{495}$

P (Subcommittee of 1 Pakistani and 3 Americans) = 0.889

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