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Chapter 23: Problem 18

A firm has 1400 employees. The probability that an employee is absent on any day is \(0.006\). Use the Poisson approximation to the binomial distribution to calculate the probability that the number of absent employees is (a) eight (b) nine

Short Answer

Expert verified
Answer: (a) The probability that exactly 8 employees are absent is about 0.179. (b) The probability that exactly 9 employees are absent is about 0.149.

Step by step solution

01

Identify the given parameters

To use the Poisson approximation, we have to identify two values: the number of employees, `n`, and the probability of an employee being absent, `p`. In the given question, n = 1400 and p = 0.006.
02

Calculate the average number of absences (parameter λ)

The average number of absences (also called the Poisson parameter λ) can be calculated by multiplying the total number of employees by the probability of each employee being absent: λ = n * p = 1400 * 0.006 = 8.4 This means, on average, 8.4 employees are absent on any given day.
03

Use the Poisson probability formula

The Poisson probability formula is given by: P(x) = (e^(-λ) * (λ^x)) / x! Where P(x) is the probability of x events occurring, λ is the average number of events occurring, and x! is the factorial of x. We'll apply this formula for x = 8 and x = 9 to find the required probabilities.
04

Calculate the probability for 8 absent employees

Using the Poisson probability formula for x = 8: P(8) = (e^(-8.4) * (8.4^8)) / 8! ≈ 0.179 The probability that exactly 8 employees are absent is about 0.179.
05

Calculate the probability for 9 absent employees

Similarly, using the Poisson probability formula for x = 9: P(9) = (e^(-8.4) * (8.4^9)) / 9! ≈ 0.149 The probability that exactly 9 employees are absent is about 0.149. The answer is: (a) The probability that exactly 8 employees are absent is about 0.179. (b) The probability that exactly 9 employees are absent is about 0.149.

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