Question

Medical billing errors and fraud are on the rise. According to Medical Billing Advocates of America, three out of four times, the medical bills that they review contain errors. If a sample of 10 medical bills is selected, what is the probability that:

A. 0 medical bills contain errors?

B. Exactly 5 medical bills contain errors?

C. More than 5 medical bills will contain errors?

D. What are the mean and standard deviation of the probability distribution?

Please show work so I can see how it is done, I looked everywhere and cannot figure this out please help!

Answer 1

X ~ Bin ( n, p )

Where n = 10 , p = 3/4 = 0.75

Binomial probability distribution is

P(X) = ⁿC_x p^X ( 1 - p)^n-X

a)

P(X = 0) = ¹⁰C₀ * 0.75⁰ * ( 1 - 0.75)¹⁰

= 0.0000

b)

P(X = 5) = ¹⁰C₅ * 0.75⁵ * ( 1 - 0.75)⁵

= 0.0584

c)

P(X > 5) = P(X >= 6)

= P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)

=  ¹⁰C₆ * 0.75⁶ * ( 1 - 0.75)⁴ + ¹⁰C₇ * 0.75⁷ * ( 1 - 0.75)³ +¹⁰C₈ * 0.75⁸ * ( 1 - 0.75)² +¹⁰C₉ * 0.75⁹ * ( 1 - 0.75)

+ ¹⁰C₁₀ * 0.75¹⁰ * ( 1 - 0.75)⁰

= 0.9219

d)

Mean = np = 10 * 0.75 = 7.5

Standard deviation = sqrt ( n p (1 - p) )

= sqrt ( 10 * 0.75 * ( 1 - 0.75) )

= 1.3693