2. In a survey of a large company, 30 percent of all employees are said that they are dissatisfied with their working conditions. Find the probabilities that among 12 randomly selected employee who were surveyed; a) None were dissatisfied with working conditions ( 4 marks) b) All were dissatisfied with working conditions ( 4marks) c) EIGHT were dissatisfied with working conditions ( 4 marks)
For a large population with a known (30%) probability of success with relatively small samples, binomial distribution may be used to model the situation.
p=0.3
n=12
P(X=x)=C(n,x)(p^x)(1-p)^(n-x)
where
C(n,x)=n!/(x!(n-x)!) combination of x objects from n.
(A) x=12
P(X=12)=C(12,12)(0.3^12)(0.7^0)
=5.3*10^(-7)
(B) x=0
P(X=0)=C(12,0)(0.3^0)(0.7^12)
=...
(C) x=exactly 8
P(X=8)=C(12,8)(0.3^8)(0.7^4)
=...
To find the probabilities in this scenario, we can use the binomial probability formula. The formula for calculating the probability of obtaining exactly k successes in n independent Bernoulli trials is:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k),
where:
- P(X = k) is the probability of obtaining precisely k successes,
- C(n, k) is the number of combinations of n objects taken k at a time (given by n! / (k!(n-k)!)),
- p is the probability of success on a single trial (in this case, the proportion of dissatisfied employees), and
- n is the number of trials (in this case, the number of employees surveyed).
a) Finding the probability that none of the 12 employees were dissatisfied with working conditions:
P(X = 0) = C(12, 0) * (0.30)^0 * (1-0.30)^(12-0)
C(12, 0) = 12! / (0!(12-0)!) = 1
P(X = 0) = 1 * 1 * 0.70^12 = 0.7^12
Calculating this, we find that the probability that none of the 12 employees were dissatisfied with their working conditions is approximately 0.0311.
b) Finding the probability that all 12 employees were dissatisfied with working conditions:
P(X = 12) = C(12, 12) * (0.30)^12 * (1-0.30)^(12-12)
C(12, 12) = 12! / (12!(12-12)!) = 1
P(X = 12) = 1 * 0.30^12 * (0.70)^0 = 0.30^12
Calculating this, we find that the probability that all 12 employees were dissatisfied with their working conditions is approximately 0.00000518.
c) Finding the probability that exactly 8 employees were dissatisfied with working conditions:
P(X = 8) = C(12, 8) * (0.30)^8 * (1-0.30)^(12-8)
C(12, 8) = 12! / (8!(12-8)!) = 495
P(X = 8) = 495 * 0.30^8 * (0.70)^4
Calculating this, we find the probability that exactly 8 employees were dissatisfied with their working conditions.