A sheriff eats donuts while checking vehicles on I75 for spe

A sheriff eats donuts while checking vehicles on I-75 for speeding. Since donuts are an important part of his diet, he will take it easy while eating. At the start of his shift, he points his radar gun at the next passing vehicle. If the vehicle is going over 8t mph, he will go after that vehicle, issues a ticket, return to his hiding post, and start over. If the vehicle is not speeding he will eat a donut to completion, then he will check the next vehicle and proceed as described previously. At that particular section of I-75, 1 of every 3 vehicles goes over 85 mph. Note: the sheriff buys a dozen donuts every day and he only checks a single vehicle at a time.

if the sheriff wants to keep a daily quota of 5 tickets, before heading home what is the probability that he will eat exactly half a dozen donuts before dinner?

A) 0.076

B) 0.189

C) 0.263

D) 0.317

Solution

If we need him to have 6 donuts, 6 vehicles dont exceed the limit and he completes 5 tickets at (6+5)th = 11th vehicle.

So, it is a Negative Binomial distribution with k = 5 and p = 1/3

P(X=11) = 10C4 * (1/3)⁵ *(2/3)⁶ = 0.076

Answer: A) 0.076