Question

You may need to use the appropriate appendix table or technology to answer this question.

A center for medical services reported that there were 295,000 appeals for hospitalization and other services. For this group, 40% of first-round appeals were successful. Suppose 10 first-round appeals have just been received by a Medicare appeals office. (Round your answers to four decimal places.)

(a)

Compute the probability that none of the appeals will be successful.

(b)

Compute the probability that exactly one of the appeals will be successful.

(c)

What is the probability that at least two of the appeals will be successful?

(d)

What is the probability that more than half of the appeals will be successful?

Answer 1

X ~ Binomial (n,p)

Where, n = 10 , p = 0.40

P(X) = ⁿC_x p^x (1-p)^n-x

a)

(X = 0) = ¹⁰C₀ 0.40⁰ 0.60¹⁰

= 0.0060

b)

P( X = 1) = ¹⁰C₁ 0.40¹ 0.60⁹

= 0.0403

c)

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

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

= 1 - [ ¹⁰C₀ 0.40⁰ 0.60¹⁰ +¹⁰C₁ 0.40¹ 0.60⁹ ]

= 0.9536

d)

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

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

= ¹⁰C₆ 0.40⁶ 0.60⁴ + ¹⁰C₇ 0.40⁷ 0.60³ +¹⁰C₈ 0.40⁸ 0.60² +¹⁰C₉ 0.40⁹ 0.60¹ +¹⁰C₁₀ 0.40¹⁰ 0.60⁰

= 0.1662