A company responds to 80% of all enquiries within 2 working days. The firm received 8 enquiries today. What is the probability that all the 8 enquiries are responded within 2 working days?
Question
A company responds to 80% of all enquiries within 2 working days. The firm received 8 enquiries today. What is the probability that all the 8 enquiries are responded within 2 working days?
Solution
To solve this problem, we can use the concept of binomial probability.
The binomial probability formula is: P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))
Where:
- P(X=k) is the probability of k successes in n trials
- C(n, k) is the combination of n items taken k at a time
- p is the probability of success
- n is the number of trials
- k is the number of successes
In this case, we want to find the probability that all 8 enquiries are responded within 2 working days. So, we are looking for P(X=8). The company responds to 80% of all enquiries within 2 working days, so p=0.8. The firm received 8 enquiries today, so n=8. And we want all enquiries to be responded, so k=8.
Substituting these values into the formula, we get:
P(X=8) = C(8, 8) * (0.8^8) * ((1-0.8)^(8-8)) = 1 * (0.8^8) * (0.2^0) = 1 * 0.16777 * 1 = 0.16777
So, the probability that all the 8 enquiries are responded within 2 working days is approximately 0.168, or 16.8%.
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