Question

Among the 300 employees of a company, 240 are union members, while the others are not. If the 8 of the employees are chosen by lot to serve on the committee which administers the pension fund, find the probability that

(a) 5 of them union members

(b) all of them union members

(c) at least 6 of them union members

(d) 2 to 4 of them union members​

Answer 1

Given : Among the 300 employees of a company, 240 are union members, while the others are not

8 of the employees are chosen by lot to serve on the committee

To Find :    probability that

(a) 5 of them union members

(b) all of them union members

(c) at least 6 of them union members

(d) 2 to 4 of them union members

Solution:

Total Employee = 300

Union members = 240   p = 240/300 = 4/5

Not union members   q  = 1 - 4/5  = 1/5

8 of the employees are chosen

n = 8

P(x) = ⁿCₓpˣqⁿ⁻ˣ

5 of them union members

x = 5

=> P(5)  = ⁸C₅(4/5)⁵(1/5)³   =  0.1468

all of them union members

x = 8

=> P(8)  = ⁸C₈(4/5)⁸(1/5)⁰   =  0.16777

at least 6 of them union members

= P(6) + P(7) + P(8)

= ⁸C₆(4/5)⁶(1/5)²  + ⁸C₇(4/5)⁷(1/5)¹  + ⁸C₈(4/5)⁸(1/5)⁰

= 0.2936 + 0.33554 + 0.16777

= 0.7969

2 to 4 of them union members​

= P(2) + P(3) + P (4)

= ⁸C₂(4/5)²(1/5)⁶ + ⁸C₃(4/5)³(1/5)⁵  + ⁸C₄(4/5)⁴(1/5)⁴

= 0.001147 + 0.009175 + 0.045875

= 0.0562

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