Harpers Index reported that the number of Orange County Cali

Harper\'s Index reported that the number of (Orange County, California) convicted drunk drivers whose sentence included a tour of the morgue was 605, of which only 1 became a repeat offender. (a) Suppose that of 966 newly convicted drunk drivers, all were required to take a tour of the morgue. Let us assume that the probability of a repeat offender is still p = 1/605. Explain why the Poisson approximation to the binomial would be a good choice for r = number of repeat offenders out of 966 convicted drunk drivers who toured the morgue. The Poisson approximation is good because n is large, p is small, and np > 10. The Poisson approximation is good because n is large, p is large, and np 1? (Use 4 decimal places.) .4634 Incorrect: Your answer is incorrect. (d) What is the probability that r > 2? (Use 4 decimal places.) (e) What is the probability that r > 3? (Use 4 decimal places.)

Solution

poisson approximation is good because n is very large here = 966 (which is way more than 100)
p is very small ( < 0.05) = 1/605 and np= 1.5967 (which is way less than 10)
so,poisson is preferred over binomial.
now, r = number of repeat offenders out of 966 convicted drunk drivers who toured the morgue.
p(r >2) = 1 - p(r <=2) = 1 - ( p(r=2)+ p(r=1)+p(r=0) )
= 1 - ( e^(-1.5967)* ( 1.5967^2) / 2! ) - ( e^(-1.5967)* ( 1.5967^1) / 1! ) - (e^(-1.5967)* ( 1.5967^0) / 0! )
=1- 0.2582 - 0.3234 - 0.2026 = 0.2158
p(r > 3 ) = 1- ( p(r=3)+ p(r=2)+ p(r=1)+p(r=0) ) = 1 - 0.1374-0.2582 - 0.3234 - 0.2026 = 0.0784