Question

11) Records provided by the vice president for human resources at a large urban hospital indicate than on any given workday 9% of the nonclinical workforce (i.e. kitchen, housekeeping and janitorial, electrical and plumbing, security, mailroom, laundry, clerical, and administrative) are absent from work. What is the probability that in a random sample of ten nonclinical workers taken on a single day: a) exactly one will be absent that day? b) at least two will be absent that day? c) none will be absent that day? d) more than three will be absent that day?

Answer 1

a)

X ~ Bin ( n,p) ,

Where n = 10 , p = 0.09

P(X) = ⁿC_x  p^x (1 - p)^n-x

a)

P(X = 1) = ¹⁰C₁ * 0.09¹ * ( 1 - 0.09)⁹

= 0.3851

b)

P(X >= 2) = 1 - P(X <= 1 )

= 1 - [ P( x= 0) + P(X = 1) ]

= 1 - [ ¹⁰C₀ * 0.09⁰ * ( 1 - 0.09)¹⁰ + ¹⁰C₁ * 0.09¹ * ( 1 - 0.09)⁹ ]

= 0.2254

c)

P(X = 0) = ¹⁰C₀ * 0.09⁰ * ( 1 - 0.09)¹⁰

= 0.3894

d)

P(X > 3) = 1 - P(X <= 3)

= 1 - [ P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) ]

= 1 - [ ¹⁰C₀ * 0.09⁰ * (1 - 0.09)¹⁰ + ¹⁰C₁ * 0.09¹ * (1 - 0.09)⁹ + ¹⁰C₂ * 0.09² * (1 - 0.09)⁸ + ¹⁰C₃ * 0.09³ * (1 - 0.09)⁷ ]

= 0.0088