50 page 1 6 IRS Prosecutions For tax evasion cases that actu

50 page 1 6 IRS Prosecutions For tax evasion cases that actually reached 5 prosecution in 2009, the IRS reported a conviction rate of5 87.2%. Suppose 20 tax evasion cases prosecuted by the IRS in 2009 are randomly selected. (a) What is the probability that all 20 resulted in convictions? (b) What is the probability that exactly 19 resulted in convictions? (c) What is the probability that at least 18 resulted in convictions? Source: Internal Revenue Service

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

a)
P( X = 20 ) = ( 20 20 ) * ( 0.872^20) * ( 1 - 0.872 )^0
= 0.0646

b)
P( X = 19 ) = ( 20 19 ) * ( 0.872^19) * ( 1 - 0.872 )^1
= 0.1897

c)
P( X = 18 ) = ( 20 18 ) * ( 0.872^18) * ( 1 - 0.872 )^2
= 0.2645

P(X>=18) = P(X=18)+P(X=19)+P(X=20) = 0.0646+0.1897+0.2645 = 0.5188